Encryption level indicator calculation method and computer program

ABSTRACT

The present invention provides a method for reliably carrying out an encryption level evaluation process in a common-key block encryption method. To be more specific, an algorithm of a key-scheduling part is expressed in terms of equations represented by vectors and a matrix, and non-linear transformation output values and initial values are eliminated from the matricial equation by carrying out a unitary transformation process in order to find all equations expressing linear relations among round keys. In accordance with the method, it is possible to comprehend all equations expressing linear relations among round keys in the common-key block encryption method without regard to the complexity of key scheduling and evaluate the encryption level of the common-key block encryption method on the basis of the derived equations expressing linear relations among round keys.

BACKGROUND OF THE INVENTION

[0001] The present invention relates to an encryption level indicatorcalculation method and a computer program. To put it in more detail, thepresent invention relates to an encryption level indicator calculationmethod for calculating an indicator for evaluating safety and level of acommon-key block encryption method as well as relates to a computerprogram implementing the encryption level indicator calculation method.

[0002] There is a variety of encryption processing algorithms, which canbe roughly classified a public key encryption method and a common-keyencryption method. The public key encryption method is an encryptionmethod, which sets an encryption key and a decryption key as differentkeys such as a public key and a private key. On the other hand, thecommon-key encryption method is an encryption method, which sets anencryption key and a decryption key as a common key.

[0003] There is also a variety of algorithms adopted in the common-keyencryption method. An encryption method adopts one of the algorithms. Inaccordance with this encryption method, a plurality of keys is generatedwith a common key used as a base and the generated keys are used incarrying out an encryption process. As a method for generating the keys,a method using a round function is adopted. To put it in detail, inaccordance with this key generation method, the round function isapplied to a common key to generate a new key on the basis of the outputvalue. Then, the round function is applied to the new key to generateanother key. Subsequently, the round function is applied to the otherkey to generate a further key. Then, the round function is applied tothe further key to generate a still further key. This procedurerepeating the operation to generate a key results in a plurality ofkeys. A representative algorithm for generating a plurality of keys asdescribed above is referred to as a common-key block encryption method.

[0004] The common-key block encryption processing algorithm can bedivided mainly into a round function part and a key-scheduling part.Conventionally, in order to secure safety against attacks related to akey or the like, an designer of encryption method is required to designa key-scheduling part with great caution in designing a common-key blockencryption method so that a simple relation among round functions is notestablished.

[0005] As an encryption method designed on the basis of such a guidingprinciple, Toshiba has proposed a common-key block encryption methodcalled Hierocrypt. For details of the Hierocrypt common-key blockencryption method, refers to, for example, a reference authored by K.Ohkuma et al. with a title of “The Block Cipher Hierocrypt,” SelectedAreas in Cryptography, LNCS 2012, pp. 72-88, 2000. The key-schedulingpart of the Hierocrypt algorithm has a repetitive structure called aFeistel structure. A linear transformation part forming the right halfof the Feistel structure tries an operation to avoid an attack relatedto a key by carrying out an XOR addition process on round-dependentconstants.

[0006] As a matter of fact, however, in the year of 2001, Furuya et al.discovered the fact that a linear relation among round keys isestablished. The fact that a linear relation among round keys isestablished was not expected by the creator of the Hierocypt algorithm.For details of the discovery made by Furuya et al., refer to, forexample, a reference authored by S. Furuya and V. Rijmen with a title of“Observations on Hierocrypt-3/L1 Key-scheduling Algorithms,” SecondNESSIE workshop, 2001.

[0007] In accordance with a method developed by Furuya et al. asdescribed in the above reference, however, an equation expressing alinear relation among round keys is derived by combining algorithms ofthe key-scheduling part of the Hierocrypt method on a trial-and-errorbasis. Thus, there is no assurance that the discovered equations are allcomprehended. In addition, with the trial-and-error basis, thedifficulty in finding a relation equation increases in case the keyscheduling becomes more complicated.

SUMMARY OF THE INVENTION

[0008] It is thus an object of the present invention addressing theproblems described above to provide an encryption level indicatorcalculation method that is capable of comprehending all linear equationsexpressing relations among round keys in a common-key block encryptionmethod without regard to the complexity of key scheduling, and capableof evaluating the encryption level of the common-key block encryptionmethod on the basis of a discovered linear-relation equation, as well asprovide a computer program implementing the encryption level indicatorcalculation method.

[0009] In accordance with a first aspect of the present invention, thereis provided an encryption level indicator calculation method based on anencryption processing algorithm and composed of:

[0010] a step of setting a common key block encryption processingalgorithm, which is to serve as the encryption processing algorithm tobe used as the base of the encryption level indicator calculationmethod, has a key-scheduling part comprising a linear transformationpart and a non-linear transformation part and includes:

[0011] a sub-step of generating initial values U_(i) (where i=1, 2 andso on) from a master key;

[0012] a sub-step of calculating intermediate values Z_(i) ⁽⁰⁾ (wherei=1, 2 and so on) from the initial values U_(i) (where i=1, 2 and soon);

[0013] a plurality of sub-steps of calculating intermediate values Z_(i)⁽⁰⁾ (where i=1, 2 and so on) from intermediate values Z_(i) ^((r-1))(where i=1, 2 and so on);

[0014] a sub-step of calculating the non-linear transformation partoutputs V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) fromthe intermediate values Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2and so on) and the initial values U_(i) (where i=1, 2 and so on); and

[0015] a sub-step of calculating round keys K_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) from the intermediate values Z_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on) and the non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on);

[0016] a step of eliminating the intermediate values Z_(i) ^((r)) (wherei=1, 2 and so on and r=1, 2 and so on) serving as variables so that theround keys K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)can be expressed as a linear combination of the initial values U_(i)(where i=1, 2 and so on) and the non-linear transformation part outputsV_(i) ^((r)) (where i=1, 2 and so on and r=1, 2. and so on);

[0017] a step of transforming the linear combination into a simultaneouslinear equation completing transposition of terms and, thus, consistingof only terms of the initial values U_(i) (where i=1, 2 and so on) andthe non-linear transformation part outputs V_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) on the right-hand side of the equation;

[0018] a step of transforming the simultaneous linear equation into amatricial equation;

[0019] a step of multiplying both the left-hand and right-hand sides ofthe matricial equation by a row-deform unitary matrix deforming a matrixon the right-hand side of the matricial equation obtained as a result oftransformation into a step matrix from the left;

[0020] a step of creating a new matrix consisting of lowest N rows of amatrix on the left-hand side of the matricial equation obtained as aresult of transformation where N is a number obtained as a result ofsubtracting the rank value of the step matrix from the number of rows inthe step matrix; and

[0021] a step of finding N linear-relation equations by multiplying acolumn vector consisting of the round keys K_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) as elements by the new matrix generatedat the preceding step,

[0022] where:

[0023] symbol U_(i) (where i=1, 2 and so on) denotes an initial value ofthe key-scheduling part;

[0024] symbol Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an intermediate value of the key-scheduling part;

[0025] symbol V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an output of the non-linear transformation part; and

[0026] symbol K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes a round key calculated from the intermediate values Z_(i) (wherei=1, 2 and so on)

[0027] In accordance with a second aspect of the present invention,there is provided a program to be executed as a computer program incarrying out an encryption level indicator calculation process based onan encryption processing algorithm and composed of:

[0028] a step of setting a common key block encryption processingalgorithm, which is to serve as the encryption processing algorithm tobe used as the base of the encryption level indicator calculation methodand includes:

[0029] a sub-step of generating initial values U_(i) (where i=1, 2 andso on) from a master key;

[0030] a sub-step of calculating intermediate values Z_(i) ⁽⁰⁾ (wherei=1, 2 and so on) from the initial values U_(i) (where i=1, 2 and soon);

[0031] a plurality of sub-steps of calculating intermediate values Z_(i)^((r)) (where i=1, 2 and so on) from intermediate values Z_(i) ^((r-1))(where i=1, 2 and so on);

[0032] a sub-step of calculating the non-linear transformation partoutputs V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) fromthe intermediate values Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2and so on) and the initial values U_(i) (where i=1, 2 and so on) ; and

[0033] a sub-step of calculating round keys K_(i) ^((r)) (where i==1, 2and so on and r=1, 2 and so on) from the intermediate values Z_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on) and the non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on);

[0034] a step of eliminating the intermediate values Z_(i) ^((r)) (wherei=1, 2 and so on and r=1, 2 and so on) serving as variables so that theround keys K_(i)(r) (where i=1, 2 and so on and r=1, 2 and so on) can beexpressed as a linear combination of the initial values U_(i) (wherei=1, 2 and so on) and the non-linear transformation part outputs V_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on);

[0035] a step of transforming the linear combination into a simultaneouslinear equation completing transposition of terms and, thus, consistingof only terms of the initial values U_(i) (where i=1, 2 and so on) andthe non-linear transformation part outputs V_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) on the right-hand side of the equation;

[0036] a step of transforming the simultaneous linear equation into amatricial equation;

[0037] a step of multiplying both the left-hand and right-hand sides ofthe matricial equation by a row-deform unitary matrix deforming a matrixon the right-hand side of the matricial equation obtained as a result oftransformation into a step matrix from the left;

[0038] a step of creating a new matrix consisting of lowest N rows of amatrix on the left-hand side of the matricial equation obtained as aresult of transformation where N is a number obtained as a result ofsubtracting the rank value of the step matrix from the number of rows inthe step matrix; and

[0039] a step of finding N linear-relation equations by multiplying acolumn vector consisting of the round keys K_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) as elements by the new matrix generatedat the preceding step,

[0040] where:

[0041] symbol U_(i) (where i=1, 2 and so on) denotes an initial value ofthe key-scheduling part;

[0042] symbol Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an intermediate value of the key-scheduling part;

[0043] symbol V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an output of the non-linear transformation part; and

[0044] symbol K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes a round key calculated from the intermediate values Z_(i) (wherei=1, 2 and so on).

[0045] In accordance with the configuration of the present invention, itis possible to comprehend all equations expressing relations among roundkeys in a common-key block encryption method without regard to thecomplexity of key scheduling and evaluate the encryption level of thecommon-key block encryption method on the basis of a discoveredlinear-relation equation.

[0046] In addition, in accordance with the configuration of the presentinvention, by expressing the key-scheduling part algorithm, which is oneof encryption processing algorithms, in terms of equations representedby vectors and a matrix and by eliminating non-linear transformationoutput values and initial values from the matrix-based equation throughuse of unitary transformation, it is possible to find alllinear-relation equations expressing relations among round keys.

[0047] It is to be noted that the computer program provided by thepresent invention is a computer program that can be presented to forexample a general-purpose computer system, which is capable of executingvarious kinds of program code, by being recorded on a recording mediumin a computer-readable form or by way of communication media such as anetwork also in a computer-readable form. Examples of the recordingmedium are a CD, an FD and an MO disc. Since the computer program ispresented to the computer system in a computer-readable form, thecomputer system is capable of carrying out a process according to theprogram.

[0048] The other objects, characteristics and merits of the presentinvention will probably become apparent from later detailed explanationsof embodiments of the present invention with reference to diagrams. Itis to be noted that the technical term ‘system’ used in thisspecification means a logical group configuration of a plurality ofapparatus, which is not necessarily put in the same case.

BRIEF DESCRIPTION OF THE DRAWING

[0049]FIG. 1 shows a flowchart referred to in explanation of anencryption level indicator calculation procedure according to thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0050] The encryption level indicator calculation method provided by thepresent invention is explained in detail as follows. First of all, anoutline of a procedure of an encryption level indicator calculationprocess is explained by referring to a flowchart shown in FIG. 1. Afterthat, embodiments implementing the encryption level indicatorcalculation process provided by the present invention are described bygiving a plurality of concrete common-key block encryption processingalgorithms as examples.

[0051] [Outline of the Encryption level Indicator Calculation Process]

[0052]FIG. 1 shows a flowchart representing the encryption levelindicator calculation process provided by the present invention. Anoutline of each processing step in the flowchart is explained asfollows.

[0053] The flowchart begins with a step S101 to set an encryptionprocessing algorithm to be used as the base of the encryption levelindicator calculation method. In this case, the encryption processingalgorithm to be used as the base of the encryption level indicatorcalculation method is a common key block encryption processingalgorithm.

[0054] To put it concretely, as the encryption processing algorithm tobe used as the base of the encryption level indicator calculationmethod, the processing at this step S101 sets a common key blockencryption processing algorithm including a key-scheduling part, whichcomprises a linear conversion part and a non-linear transformation part,and having:

[0055] a step of generating initial values U_(i) (where i=1, 2 and soon) from a master key;

[0056] a step of calculating intermediate values Z_(i) ⁽⁰⁾ (where i=1, 2and so on) from the initial values U_(i) (where i=1, 2 and so on);

[0057] a plurality of steps of calculating intermediate values Z_(i)^((r)) (where i=1, 2 and so on) from intermediate values Z_(i) ^((r-l))(where i=1, 2 and so on);

[0058] a step of calculating the non-linear transformation part outputsV_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) from theintermediate values Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 andso on) and the initial values U_(i) (where i=1, 2 and so on) ; and

[0059] a step of calculating round keys K_(i) ^((r)) (where i=1, 2 andso on and r=1, 2 and so on) from the intermediate values Z_(i) ^((r))(where i=1, 2 and so on and r=1, 2 and so on) and the non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on), where:

[0060] symbol U_(i) (where i=1, 2 and so on) denotes an initial value ofthe key-scheduling part;

[0061] symbol Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an intermediate values of the key-scheduling part;

[0062] symbol V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes an output of the non-linear transformation part; and

[0063] symbol K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on)denotes a round key calculated from the intermediate values Z_(i) (wherei=1, 2 and so on).

[0064] Then, at the next step S102, intermediate variables of thecommon-key block encryption processing algorithm set at the step S101are eliminated. To put it concretely, the processing carried out at thestep S102 eliminates the intermediate values Z_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) so that the round keys K_(i) ^((r))(where i=1, 2 and so on and r=1, 2 and so on) can be expressed as alinear combination of the initial values U_(i) (where i=1, 2 and so on)and the non-linear transformation part outputs V_(i) ^((r)) (where i=1,2 and so on and r=1, 2 and so on). The concrete example of theprocessing will be described later.

[0065] Then, at the next step S103, a variable transposition process iscarried out. To put it concretely, the processing carried out at thestep S103 transforms the expression of the linear combination into asimultaneous linear equation completing transposition of terms and,thus, consisting of only terms of the initial values U_(i) (where i=1, 2and so on) and the non-linear transformation part outputs V_(i) ^((r))(where i=1, 2 and so on and r=1, 2 and so on). The concrete example ofthe processing will be described later.

[0066] Then, at the next step S104, a matricial-equation transformationprocess is carried out. The matricial-equation transformation process isa process to transform the simultaneous linear equation into a matricialequation. The matricial-equation transformation process will beexplained in concrete terms later.

[0067] Then, at the next step S105, a unitary transformation process iscarried out. To put it in detail, both the left-hand and right-handsides of the matricial equation are multiplied by a row-deform unitarymatrix deforming a matrix on the right-hand side of the matricialequation obtained as a result of transformation into a step matrix fromthe left. An embodiment of the unitary transformation process will bedescribed later.

[0068] Then, at the next step S106, a small-matrix selection process iscarried out. To put it in detail, the small-matrix selection process isa process to create a new matrix consisting of lowest N rows of a matrixon the left-hand side of the matricial equation obtained as a result oftransformation where N is a number obtained as a result of subtractingthe rank value of the step matrix from the number of rows in the stepmatrix. An embodiment of the small-matrix selection process will bedescribed later.

[0069] Then, at the next step S107, a linear-relation equationgeneration process is carried out. To put it in detail, thelinear-relation equation generation process is a process to find Nlinear-relation equations by multiplying a column vector consisting ofthe round keys K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and soon) as elements by the new matrix generated at the preceding step S106.An embodiment of the linear-relation equation generation process will bedescribed later.

[0070] The number (N) of linear-relation equations found in the processcarried out at the step S107 is the encryption level indicator of thecommon-key block encryption algorithm set at the step S101. Theprocessing represented by the flowchart described above is executed as aprocess to find the value of N, which is number of linear-relationequations comprehensively including equations representing linearrelations among round keys of the common-key block encryption algorithmset at the step S101. The larger the number (N) of linear-relationequations, the smaller the encryption level. Conversely speaking, thesmaller the number (N) of linear-relation equations, the larger theencryption level. Thus, the number (N) of linear-relation equationsfound by carrying out the processing represented by the flowchart shownin FIG. 1 can be used as the encryption level indicator of thecommon-key block encryption algorithm.

[0071] In accordance with the processing according to the processingprocedure represented by the flowchart shown in FIG. 1, thekey-scheduling part algorithm, which is one of encryption algorithms, isexpressed by a matricial equation represented by vectors and a matrix.By eliminating non-linear transformation output values and initialvalues from the matricial equation through a unitary transformationprocess, all equations of linear relations among round keys can befound.

[0072] [First Embodiment of Encryption Level Indicator CalculationProcess]

[0073] As a first embodiment of the encryption level indicatorcalculation process provided by the present invention, a typical processof applying an encryption level evaluation method provided by thepresent invention to ‘Hierocrypt-L1’ is explained in detail.‘Hierocrypt-L1’ is the name of a block encryption process proposed byToshiba. The ‘Heirocrypt-L1’ block encryption process is a common-keyblock encryption process with a block length of 64 bits and a key lengthof 128 bits.

[0074] First of all, the step S101 of the flowchart shown in FIG. 1 isexplained. As described earlier, at this step, an encryption processingalgorithm is set. This step is executed as a process to set the‘Hierocrypt-L1’ block encryption algorithm proposed by Toshiba.

[0075] Let symbol On denote a null matrix consisting of n rows and ncolumns whereas symbol In denote a unit matrix consisting of n rows andn columns. In this case, a matrix P16 is defined as follows:

[0076] [Formula 1] ${P16} = \begin{pmatrix}{I2} & {O2} & {I2} & {O2} \\{O2} & {I2} & {O2} & {I2} \\{O2} & {I2} & {I2} & {I2} \\{I2} & {O2} & {I2} & {I2}\end{pmatrix}$

[0077] Let symbol P16I denote the inverse matrix of the matrix P16.Next, matrices M5 and MB are defined as follows: $\begin{matrix}{{M5} = \begin{pmatrix}1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\0 & 1 & 0 & 1\end{pmatrix}} \\{{M\quad B} = \begin{pmatrix}0 & 1 & 0 & 1 \\1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 0 & 1 & 1\end{pmatrix}}\end{matrix}$

[0078] Then matrices M5B and MB5 are defined, being expressed in termsof the matrices M5 and MB as follows:

[0079] [Formula 3] $\begin{matrix}{{M5B} = \begin{pmatrix}{M5} & {O4} \\{O4} & {M\quad B}\end{pmatrix}} \\{{M\quad {B5}} = \begin{pmatrix}{M\quad B} & {O4} \\{O4} & {M5}\end{pmatrix}}\end{matrix}$

[0080] Next, round dependent constant vectors Gi (where i=0, . . . , 7)are defined as follows:

G0=(h01, h02, h03, h04, 0, 0, 0, 0)

G1=(h11, h12, h13, h14, 0, 0, 0, 0)

G2=(h21, h22, h23, h24, 0, 0, 0, 0)

G3=(h31, h32, h33, h34, 0, 0, 0, 0)

G4=(h41, h42, h43, h44, 0, 0, 0, 0)

G5=(h41, h42, h43, h44, 0, 0, 0, 0)

G6=(h31, h32, h33, h34, 0, 0, 0, 0)

G7=(h21, h22, h23, h24, 0, 0, 0, 0)   [formula 4]

[0081] It is to be noted that a vector HH with constants in the aboveequations used as elements is defined as follows:

HH=(h01, h02, h03, h04, h11, h12, h13, h14, h21, h22, h23, h24, h31,h32, h33, h34, h41, h42, h43, h44,)   [formula 5]

[0082] The actual values of the elements h01, h02, . . . and h44 aredefined as follows.

(h01, h01, h02, h03)=(0×5a, 0×82, 0×79, 0×99)

(h11, h11, h12, h13)=(0×6e, 0×9, 0×eb, 0×xz1)

(h21, h21, h22, h23)=(0×8f, 0×1b, 0×bc, 0×dc)

(h31, h31, h32, h33)=(0×ca, 0×62, 0×c1, 0×d6)

(h41, h42, h43, h44)=(0×xf7, 0×de, 0×f5, 0×8a)   [formula 6]

[0083] Next, a vector ZZ with its elements composing the right half of asequence of initial values of the key-scheduling part is defined asfollows.

ZZ=(z31, z32, z33, z34, z41, z42, z43, z44)   [formula 7]

[0084] By using these, the right half of the key-scheduling part in theHierocrypt-L1 common-key encryption algorithm is expressed below. It isto be noted that the operator + used in the following expressions is anadditive operator in the Galois field GF(2).

Z0=M5B*ZZ+G0

W0=P16*Z0

Z1=M5B*W0+G1

W1=P16*Z1

Z2=M5B*W1+G2

W2=P16*Z2

Z3=M5B*W2+G4

W3=P16*Z3

Z4=M5B*W3+G4

W5=M5B*(Z4+G5)

Z5=P16I*W5

W6=M5B*(Z5+G6)

Z6=P16I*W6

W7=M5B*(Z6+G7)

Z7=P16I*W7   [formula 8]

[0085] Symbols Z0, Z1, Z2, Z3, Z4, Z5, Z6, Z7, WO, W1, W2, W3, W5, W6and W7 used in the above equations form the right half of the sequenceof intermediate values of the key-scheduling part.

[0086] Next, these intermediate values are expressed by being split inaccordance with the following equations.

Zn=Zn ₃ ||ZN ₄

Wn=Wn ₁ ||WN ₂   [formula 9]

[0087] Symbol || used in the above equations denotes a concatenation ofvectors.

[0088] Next, let non-linear transformation part outputs of rounds be V0,V1, V2, V3, V4, V5, V6 and V7. Each of the outputs is a vectorconsisting of four elements as follows.

V0=(v01, v02, v03, v04)

V1=(v11, v12, v13, v14)

V2=(v21, v22, v23, v24)

V3=(v31, v32, v33, v34)

V4=(v41, v42, v43, v44)

V5=(v51, v52, v53, v54)

V6=(v61, v62, v63, v64)

V7=(v71, v72, v73, v74)   [formula 10]

[0089] Here, vectors Z₁ and Z₂ are set as follows.

Z₁=(z11, z12, z13, z14)

Z₂=(z21, z22, z23, z24)   [Formula 11]

[0090] With the vectors Z₁ and Z₂ set as described above, the left halfof the key-scheduling part in the Hierocrypt-L1 common-key encryptionalgorithm is expressed as follows.

Z0₁=Z₂

Z0₂ =Z ₁ +V0

Z1₁=Z0₂

Z1₂ =Z0₁ +V1

Z2₁=Z1₂

Z2₂ =Z1₁ +V2

Z3₁=Z2₂

Z3₂ =Z2₁ +V3

Z4₁=Z3₂

Z4₂ =Z3₁ +V4

Z5₁ =Z4₂ +V5

Z5₂=Z4₁

Z6₁ =Z5₂ +V6

Z6₂=Z5₁

Z7₁ =Z5₂ +V7

Z7₂=Z6₁   [formula 12]

[0091] Symbols Z0₁, Z0₂, Z1₁, Z1₂, Z2₁, Z2₂, Z3₁, Z3₂, Z4₁, Z4₂, Z5₁,Z5₂, Z6₁, Z6₂, Z7₁ and Z7₂ used in the above equations form the lefthalf of the sequence of intermediate values of the key-scheduling part.

[0092] By using the intermediate values obtained as described above,round keys K1₁, K1₂, K1₃, K1₄t K2₁, . . . , K7₁ and K7₂ are expressed asfollows:

K1₁ =Z0₁ +V1

K1₂ =Z1₃ +V1

K1₃ =Z1₄ +V1

K1₄ =Z0₂ +z1₄

K2₁ =Z1₁ +V2

K2₂ =Z2₃ +V2

K2₃ =Z2₄ +V2

K2₄ =Z1₂ +Z2₄

K3₁ =Z2₁ +V3

K3₂ =Z3₃ +V3

K3₃ =Z3₄ +V3

K3₄ =Z2₂ +Z3₄

K4₁ =Z3₁ +V4

K4₂ =Z4₃ +V4

K4₃ =Z4₄ +V4

K5₁ =Z5₁ +Z4₃

K5₂ =W5₁ +V5

K5₃ =W5₂ +V5

K5₄ =Z4₁ +W5₂

K6₁ =Z6₁ +Z5₃

K6₂ =W6₁ +V6

K6₃ =W6₂ +V6

K6₄ =Z5₁ +W6₂

K7₁ =Z7₁ +Z6₃

K7₂ =W7₁ +V7

K7₃ =W7₂ +V7

K7₄ =Z6₁ +W7₂   [formula 13]

[0093] It is to be noted that symbols K1₁, K1₂, K1₃, K1₄, K2₁, . . . ,K7₁ and K7₂ each denote a vector consisting of four elements.

[0094] The following description explains the step S102 of carrying outa process to eliminate intermediate variables in the processingrepresented by the flowchart shown in FIG. 1. If the four elements ofeach of the vectors. K1₁, K1₂, K1₃, K1₄, K2₁, . . . , K7₁ and K7₂ areexpressed by their actual values, the vectors K1₁, K1₂, K1₃, K1₄, K2₁, .. . , K7₁ and K7₂ can be expressed as follows:

[0095] [Formula 14] $\begin{matrix}{{K1}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{v11} + {z21}} \\{{v12} + {z22}}\end{matrix} \\{{v13} + {z23}}\end{matrix} \\{{v14} + {z24}}\end{pmatrix}} \\{{K1}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {v11} + {z32} + {z41}} \\{{h01} + {h02} + {h12} + {h04} + {v12} + {z33} + {z42}}\end{matrix} \\{{h01} + {h02} + {h03} + {h13} + {v13} + {z31} + {z34} + {z43}}\end{matrix} \\{{h02} + {h04} + {h14} + {v14} + {z31} + {z44}}\end{pmatrix}} \\{{K1}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h02} + {h04} + {v11} + {z31}} \\{{h01} + {h03} + {v12} + {z32}}\end{matrix} \\{{h02} + {h03} + {h04} + {v13} + {z32} + {z41} + {z33}}\end{matrix} \\{{h01} + {h02} + {h03} + {v14} + {z31} + {z34} + {z44}}\end{pmatrix}} \\{{K1}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h02} + {h04} + {v01} + {z11} + {z31}} \\{{h01} + {h03} + {v02} + {z12} + {z32}}\end{matrix} \\{{h02} + {h03} + {h04} + {v03} + {z13} + {z32} + {z41} + {z33}}\end{matrix} \\{{h01} + {h02} + {h03} + {v04} + {z31} + {z14} + {z34} + {z44}}\end{pmatrix}} \\{{K2}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{v01} + {v21} + {z11}} \\{{v02} + {v22} + {z12}}\end{matrix} \\{{v03} + {v23} + {z13}}\end{matrix} \\{{v04} + {v24} + {z14}}\end{pmatrix}} \\{{K2}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h02} + {h11} + {h03} + {h21} + {h13} + {v21} + {z33} + {z34} + {z43}} \\{{h11} + {h03} + {h12} + {h04} + {h22} + {h14} + {v22} + {z31} + {z41} +} \\{{z33} + {z42} + {z34}}\end{matrix} \\{{h11} + {h12} + {h04} + {h13} + {h23} + {v23} + {z32} + {z42} + {z34} + {z43}}\end{matrix} \\{{h01} + {h02} + {h12} + {h14} + {h\quad 24} + {v24} + {z32} + {z33} + {z42} + {z34}}\end{pmatrix}} \\{{K2}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h12} + {h14} + {v21} + {z31} + {z33} + {z42} + {z44}} \\{{h02} + {h11} + {h13} + {v22} + {z31} + {z32} + {z41} + {z34} + {z43}}\end{matrix} \\{{h02} + {h12} + {h13} + {h14} + {v23} + {z31} + {z32} + {z41} + {z42} +} \\{{z34} + {z43} + {z44}}\end{matrix} \\{{h01} + {h02} + {h11} + {h12} + {h04} + {h13} + {v24} + {z41} + {z33} +} \\{{z42} + {z43} + {z44}}\end{pmatrix}} \\{{K2}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h12} + {h14} + {v11} + {z21} + {z31} + {z33} + {z42} + {z44}} \\{{h02} + {h11} + {h13} + {v12} + {z22} + {z31} + {z32} + {z41} + {z34} + {z43}}\end{matrix} \\{{h02} + {h12} + {h13} + {h14} + {v13} + {z31} + {z23} + {z32} + {z41} +} \\{{z42} + {z34} + {z43} + {z44}}\end{matrix} \\{{h01} + {h02} + {h11} + {h12} + {h04} + {h13} + {v14} + {z41} + {z24} +} \\{{z33} + {z42} + {z43} + {z44}}\end{pmatrix}} \\{{K3}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{v11} + {v31} + {z21}} \\{{v12} + {v32} + {z22}}\end{matrix} \\{{v13} + {v33} + {z23}}\end{matrix} \\{{v14} + {v34} + \quad 24}\end{pmatrix}} \\{{K3}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h03} + {h12} + {h21} + {h04} + {h13} + {h31} + {h23} + {v31} +} \\{{z41} + {z42} + {z34} + {z43}} \\{{h01} + {h21} + {h13} + {h22} + {h14} + {h32} + {h24} + {v32} + {z31} +} \\{{z41} + {z33} + {z43}}\end{matrix} \\{{h01} + {h02} + {h21} + {h22} + {h14} + {h23} + {h33} + {v33} + {z32} +} \\{{z41} + {z33} + {z34}}\end{matrix} \\{{h02} + {h11} + {h03} + {h12} + {h22} + {h24} + {h34} + {v34} + {z41} + {z33} +} \\{{z42} + {z34} + {z44}}\end{pmatrix}} \\{{K3}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h03} + {h04} + {h22} + {h24} + {v31} + {z31} +} \\{{z32} + {z41}} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h23} + {v32} + {z32} + {z33} + {z42}}\end{matrix} \\{{h03} + {h12} + {h22} + {h23} + {h24} + {v33} + {z31} + {z32} + {z33} + {z42}}\end{matrix} \\{{h01} + {h02} + {h11} + {h12} + {h21} + {h04} + {h22} + {h14} + {h23} +} \\{{v34} + {z41} + {z33} + {z42} + {z44}}\end{pmatrix}} \\{{K3}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + \quad {h11} + {h03} + {h04} + {h22} + {h24} + {v01} + {v21} +} \\{{z11} + {z31} + {z32} + {z41}} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h23} + {v02} + {v22} + {z12} +} \\{{z32} + {z33} + {z42}}\end{matrix} \\{\quad {{h03} + {h12} + {h22} + {h23} + \quad {h24} + {v03} + {v23} + {z13} + {z31} +}} \\{{z32} + {z33} + {z42}}\end{matrix} \\{{h01} + {h02} + {h11} + {h12} + {h21} + {h04} + {h22} + {h14} + {h23} +} \\{{{v04} + {v24} + {z14} + {z41} + \quad {z33} + {z42} + {z44}}\quad}\end{pmatrix}} \\{{k4}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{v01} + {v21} + {v41} + {z11}} \\{{v02} + {v22} + {v42} + {z12}}\end{matrix} \\{{v03} + {v23} + {v43} + {z13}}\end{matrix} \\{{v04} + {v24} + {v44} + {z14}}\end{pmatrix}} \\{{k4}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {h13} + {h22} + {h31} + {h14} + {h23} + {h41} +} \\{{h33} + {v41} + {z32} + {z41} + {z43}} \\{{h11} + {h31} + {h23} + {h32} + {h24} + {h42} + {h34} + {v42} + {z41}}\end{matrix} \\{{h02} + {h11} + \quad {h12} + {h04} + {h31} + {h32} + {h24} + {h33} + {h43} +} \\{{v43} + {z31} + {z42}}\end{matrix} \\{{h02} + {h12} + {h21} + {h13} + {h22} + {h32} + {h34} + {h44} + {v44} +} \\{{z31} + {z32} + {z41} + {z42} + {z34} + {z43}}\end{pmatrix}} \\{{k4}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h03} + {h12} + {h21} + {h13} + {h14} + {h32} +} \\{{h34} + {v41} + {z31} + {z42} + {z34} + {z43} + {z44}} \\{{h02} + {h03} + {h12} + {h04} + {h13} + {h22} + {h31} + {h14} + {h33} +} \\{{v42} + {z32} + {z33} + {z42} + {z43}}\end{matrix} \\{{h01} + {h02} + {h03} + {h04} + {h13} + {h22} + {h32} + {h33} + {h34} +} \\{{v43} + {z31} + {z32} + {z43} + {z44}}\end{matrix} \\{{h02} + {h11} + {h12} + {h21} + {h04} + {h22} + {h31} + {h14} + {h32} +} \\{{h24} + {h33} + {v44} + {z31} + {z42} + {z44}}\end{pmatrix}} \\{{k4}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h\quad 03} + {h\quad 12} + {h21} + {h13} + {h14} + {h32} +} \\{{h34} + {v11} + {v31} + {z21} + {z31} + {z42} + {z34} + {z43} + {z44}} \\{{h02} + {h03} + {h12} + {h04} + {h13} + {h22} + {h31} + {h14} + {h33} +} \\{{v12} + {v32} + {z22} + {z32} + {z33} + {z42} + {z43}}\end{matrix} \\{{h01} + {h02} + {h03} + {h04} + {h13} + {h22} + {h32} + {h33} + {h34} +} \\{{v13} + {v33} + {z31} + {z23} + {z32} + {z43} + {z44}}\end{matrix} \\{{h02} + {h11} + {h12} + {h21} + {h04} + {h22} + {h31} + {h14} + {h32} +} \\{{h24} + {h33} + {v14} + {v34} + {z31} + {z24} + {z42} + {z44}}\end{pmatrix}} \\{{K5}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {h13} + {h22} + {h31} + {h14} + {h23} + {h41} +} \\{{h33} + {v01} + {v21} + {v41} + {v51} + {z11} + {z32} + {z41} + {z43}} \\{{h11} + {h31} + {h23} + {h32} + {h24} + {h42} + {h34} + {v02} + {v22} +} \\{{v42} + {v52} + {z12} + {z41}}\end{matrix} \\{{h02} + {h11} + {h12} + {h04} + {h31} + {h32} + {h24} + {h33} + {h43} +} \\{{v03} + {v23} + {v43} + {v53} + {z13} + {z31} + {z42}}\end{matrix} \\{{h02} + {h12} + {h21} + {h13} + {h22} + {h32} + {h34} + {h44} + {v04} +} \\{{v24} + {v44} + {z31} + {v54} + {z14} + {z32} + \quad {z41} + {z42} + \quad {z34} + {z43}}\end{pmatrix}} \\{{K5}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h02} + {h11} + {h12} + {h21} + {h13} + {h22} + {h31} + {h23} + {h24} +} \\{{v51} + {z31} + {z32} + {z42} + {z34} + {z43}} \\{{h01} + {h02} + {h03} + {h12} + {h04} + {h13} + {h22} + {h14} + {h23} +} \\{{h32} + {h24} + {v52} + {z31} + {z32} + {z41} + {z42} + {z43}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h14} + {h24} + {h33} + {v53} +} \\{{z31} + {z41} + {z42} + {z34}}\end{matrix} \\{{h01} + {h03} + {h21} + {h04} + {h14} + {h23} + {h24} + {h34} +} \\{{v54} + {z34}}\end{pmatrix}} \\{{K5}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h11} + {h12} + {h21} + {h04} + {h22} + {h14} + {h33} + {v51} + {z32} +} \\{{z42} + {z34}} \\{{h01} + {h02} + {h12} + {h14} + {h24} + {h34} + {v52} + {z32} + {z33} +} \\{{z42} + {z34}}\end{matrix} \\{{h02} + {h11} + {h03} + {h21} + {h13} + {h31} + {v53} + {z33} + {z34} + {z43}}\end{matrix} \\{{h11} + {h03} + {h21} + {h13} + {h32} + {h24} + {z31} + {v54} + {z32} +} \\{{z33} + {z43} + {z44}}\end{pmatrix}} \\{{K5}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h11} + {h12} + {h21} + {h04} + {h22} + {h14} + {h33} + {v11} + {v31} +} \\{{z21} + {z32} + {z42} + {z34}} \\{{h01} + {h02} + {h12} + {h14} + {h24} + {h34} + {v12} + {v32} + {z22} + {z32} +} \\{{z33} + {z42} + {z34}}\end{matrix} \\{{h02} + {h11} + {h03} + {h21} + {h13} + {h31} + {v13} + {v33} + {z23} +} \\{{z33} + {z34} + {z43}}\end{matrix} \\{{h11} + {h03} + {h21} + {h13} + {h32} + {h24} + {v14} + {v34} + {z31} +} \\{{z32} + {z24} + {z33} + {z43} + {z44}}\end{pmatrix}} \\{{K6}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h03} + {h12} + {h21} + {h04} + {h13} + {h31} + {h23} + {v11} +} \\{{v31} + {v61} + {z21} + {z41} + {z42} + {z34} + {z43}} \\{{h01} + {h21} + {h13} + {h22} + {h14} + {h32} + {h24} + {v12} + {v32} +} \\{{v62} + {z22} + {z31} + {z41} + {z33} + {z43}}\end{matrix} \\{{h01} + {h02} + {h21} + {h22} + {h14} + {h23} + {h33} + {v13} + {v33} +} \\{{v63} + {z23} + {z32} + {z41} + {z33} + {z34}}\end{matrix} \\{{h02} + {h11} + {h03} + {h12} + {h22} + {h24} + {h34} + {v14} + {v34} +} \\{{z41} + {v64} + {z24} + {z\quad 33} + {z42} + {z34} + {z44}}\end{pmatrix}} \\{{K6}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h03} + {h12} + {h13} + {h14} + {v61} + {z31} +} \\{{z42} + {z34} + {z43} + {z44}} \\{{h02} + {h03} + {h12} + {h04} + {h13} + {h22} + {h14} + {v62} + {z32} +} \\{{z33} + {z42} + {z43}}\end{matrix} \\{{h02} + {h11} + {h04} + {h14} + {h23} + {z31} + {v63} + {z41} + {z44}}\end{matrix} \\{{h11} + {h04} + {h13} + {h14} + {h24} + {h32} + {z41} + {v64} +} \\{{z34} + {z43} + {z44}}\end{pmatrix}} \\{{K6}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}{\quad {{h01} + {h02} + {h11} + {h12} + {h04} + {h\quad 23} + {v61} + {z41} + \quad {z33} + {z42}}} \\{{h02} + {h04} + {h14} + {h24} + {v62} + {z31} + {z44}}\end{matrix} \\{\quad {{h01} + {h11} + {h03} + {h21} + {v63} + {z32} + {z41}}}\end{matrix} \\{{h01} + {h11} + {h03} + {h22} + {h14} + {z32} + {z41} + {v64} + {z44}}\end{pmatrix}} \\{{K6}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h12} + {h04} + {h23} + {v01} + {v21} + {v41} +} \\{{v51} + {z11} + {z41} + {z33} + {z42}} \\{{h02} + {h04} + {h14} + {h24} + {v02} + {v22} + {v42} + {v52} +} \\{{z12} + {z31} + {z44}}\end{matrix} \\{{h01} + {h11} + {h03} + {h21} + {v03} + {v23} + {v43} + {v53} +} \\{{z13} + {z32} + {z41}}\end{matrix} \\{{h01} + {h11} + {h03} + {h22} + {h14} + {v04} + {v24} + {v44} +} \\{{v54} + {z14} + {z32} + {z41} + {z44}}\end{pmatrix}} \\{{K7}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}{{h02} + {h11} + {h03} + {h21} + {h13} + {v01} + {v21} + {v41} + {v51} +} \\{{z11} + {v71} + {z33} + {z34} + {z43}} \\{{h11} + {h03} + {h12} + {h04} + {h22} + {h14} + {v02} + {v22} + {v42} +} \\{{v52} + \quad {z12} + {z31} + {v72} + {z41} + {z33} + {z42} + {z34}}\end{matrix} \\{{h11} + \quad {h12} + \quad {h04} + {h13} + {h23} + {v03} + {v23} + {v43} + {v53} +} \\{{z13} + {z32} + {v73} + {z42} + {z34} + {z43}}\end{matrix} \\{{h01} + {h02} + {h12} + {h14} + {h24} + {v04} + {v24} + {v44} + {v54} +} \\{{z14} + {z32} + {z33} + {z42} + {z44} + {v74} + {z34}}\end{pmatrix}} \\{{K7}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}{\quad {{h01} + {h02} + {h11} + {h03} + {h04} + {v71} + {z31} + {z32} + {z41}}} \\{{h02} + {h03} + {h12} + {h04} + {v72} + {z32} + {z33} + {z42}}\end{matrix} \\{{h01} + {h04} + {h13} + {z31} + {z32} + {z41} + {v73} + {z33} + {z34} + {z43}}\end{matrix} \\{{h01} + {h03} + {h04} + {h14} + {v74} + {z34}}\end{pmatrix}}\end{matrix}$

[0096] Then, the next step S103 is executed to carry out a variabletransposition process. With the results of the vectors K1₁, K1₂, K1₃,K1₄, K2₁, . . . , K7₁ and K7₂ used as a base, the simultaneous linearequation is transformed so as to result in equations, which each includeonly terms zxx and vxx on the right-hand side thereof as follows.

k1₁₁ =v11+z21

k1₁₂ =v12+z22

k1₁₃ =v13+z23

k1₁₄ =v14+z24

h01+h11+h03k1₂₁ =v11z32+z41

h01+h02+h12+h04+k1₂₂ =v12+z33+z42

h01+h02+h03+h13+k1₂₃ =v13+z31+z34+z43

h02+h04+h14+k1₂₄ =v14+z31+z44

h02+h04+k1₃₁ =v11+z31

h01+h03+k1₃₂ =v12+z32

h02+h03+h04+k1₃₃ =v13+z32+z41+z33

h01+h02+h03+k1₃₄ =v14+z31+z34+z44

h02+h04+k1₄₁ =v01+z11+z31

h01+h03+k1₄₂ =v02+z12+z32

h02+h03+h04+k1₄₃ =v03+z13+z32+z41+z33

h02+h02+h03+k1₄₄ =v04+z31+z14+z34+z44

k2₁₁ =v01+v21+z11

k2₁₂ =v02+v22+z12

k2₁₃ =v03+v23+z13

k2₁₄ =v04+v24+z14

h02+h11h03+h21+h13+k2₂₁ =v21+v33+z34+z43

h11+h03+h12+h04+h22+h14+k2₂₂ =v22+z31+z41+z33z42+z34

h11+h12+h04+h13+h23+k2₂₃ =v23+z32+z42+z34+z43

h01+h02+h12+h14+h24+k2₂₄ =v24+z32+z33+z42+z34

h01+h12+h14+k2₃₁ =v21+z31+z33+z42+z44

h02+h11+h13+k2₃₂ =v22+z31+z32+z41+z34+z43

h02+h12+h13+h14+k2₃₃ =v23+z31+z32+z41+z42+z34+z43+z44

h01+h02+h11+h12+h04+h13+k2₃₄ =v24+z41+z33+z42+z43+z44

h01+h12+h14+k2₄₁ =v11+z21+z31+z33+z42+z44

h02+h11+h13+k2₄₂ =v12+z22+z31+z32+z41+z34+z43

h02+h12+h13+h14+k2₄₃ =v13+z31+z23+z32+z41+z42+z34+z42+z44

h01+h02+h11+h12+h04+h13+k2₄₄ =v14+z41+z24+z33+z42+z43+z44

k3₁₁ =v11+v31+z21

k3₁₂ =v12+v32+z22

k3₁₃ =v13+v33+z23

k3₁₄ =v14+v34+z24

h01+h03+h12+h21+h04+h13+h31+h23+k3₂₁ =v31+z41+z42+z34+z43

h01+h21+h13+h22+h14+h32+h24+k3₂₂ =v32+z31+z41+z33+z43

h01+h02+h21+h22+h14+h23+h33+k3₂₃ =v33+z32+z41+z33+z34

h02+h11+h03+h12+h22+h24+h34+k3₂₄ =v34+z41+z33+z42+z34+z44

h01+h02+h11+h03+h04+h22+h24+k3₃₁ =v31+z31+z32+z41

h02+h03+h12+h21+h04+h23+k3₃₂ =v32+z32+z33+z42

h03+h12+h22+h23+h24+k3₃₃ =v33+z31+z32+z33+z42

h01+h02+h11+h12+h21+h04+h22+h14+h23+k3₃₄ =v34+z41+z33+z42+z44

h01+h02+h11+h03+h04+h22+h24+k3₄₁ =v01+v21+z11+z31+z32+z41

h02+h03+h12+h21h04+h23+k3₄₂ =v02+v22+z12+z32+z33+z42

h03+h12+h22+h23+h24+k3₄₃ =v03+v23+z13+z31+z32+z33+z42

h01+h02+h11+h12+h21+h04+h22+h14+h23+k3₄₄ =v04+v24+z14+z41z33+z42+z44

k4₁₁ =v01+v21+v41+z11

k4₁₂ =v02+v22+v42+z12

k4₁₃ =v03+v23+v43+z13

k4₁₄ =v04+v24+v44+z14

h01+h11+h03+h13+h22+h31+h14+h23+h41+h33+k4₂₁ =v41+z32+z41+z43

h11+h31+h23+h32+h24+h42+h34+k4₂₂ =v42+z41

h02+h11+h12+h04+h31+h32+h24+h33+h43+k4₂₃ =v43+z31+z42

h02+h12+h21+h13+h22+h32+h34+h44+k4₂₄ =v44+z31+z32+z41+z42+z34+z43

h01+h02+h11+h03+h12+h21+h13+h14+h32+h34+k4₃₁ =v41+z31+z42+z34+z43+z44

h02+h03+h12+h04+h13+h22+h31+h14+h33+k4₃₂ =v42+z32+z33+z42+z43

h01+h02+h03+h04+h13+h22+h32+h33+h34+k4₃₃ =v43+z31+z32+z43+z44

h02+h11+h12+h21+h04+h22+h31+h14+h32+h24+h33+k4₃₂ =v44+z31+z42+z44

h01+h02+h11+h03+h12+h21+h13+h14+h32+h34+k4₄₁=v11+v31+z21+z31+z42+z34+z43+z44

h02+h03+h12+h04+h13+h22+h31+h14+h33+k4₄₂ =v12+v32+z22+z32+z33+z42+z43

h01+h02+h03+h04+h13+h22+h32+h33+h34+k4₄₃ =v13+v33+z31+z23+z32+z43+z44

h02+h11+h12+h21+h04+h22+h31+h14+h32+h24+h33+k4₄₄=v14+v34+z31+z24+z42+z44

h01+h11+h03+h13+h22+h31+h14+h23+h41+h33+k5₁₁=v01+v21+v41+v51+z11+z32+z41+z43

h11+h31+h23+h32+h24+h42+h34+k5₁₂ =v02+v22+v42+v52+z12+z41

h02+h11+h12+h04+h31+h32+h24+h33+h43+k5₁₃ =v03+v23+v43+v53+z13+z31+z42

h02+h12+h21+h13+h22+h32+h34+h44+k5₁₄=v04+v24+v44+v54+z14+z32+z41+z42+z34+z43

h02+h11+h12+h21+h13+h22+h31+h23+h24+k5₂₁ =v51+z31+z32+z42+z34+z43

h01+h02+h03+h12+h04+h13+h22+h14+h23+h32+h24+k5₂₂ =v52+z31+z32+z42+z43

h01+h02+h03+h12+h21+h14+h24+h33+k5₂₃ =v53+z31+z41+z42+z34

h01+h03+h21+h04+h14+h23+h24+h34+k5₂₄ =v54+z34

h11+h12+h21+h04+h22+h14+h33+k5₃₁ =v51+z32+z42+z34

h01+h02+h12+h14+h24+h34k5₃₂ =v52+z32+z33+z42+z34

h02+h11+h03+h21+h13+h31k5₃₃ =v53+z33+z34+z43

h11+h03+h21+h13+h32+h24+z31+k5₃₄ =v54+z32+z33+z43+z44

h11+h12+h21+h04+h22+h14+h33k5₄₁ =v11+v31+z21+z32+z42+z34

h01+h02+h12+h14+h24+h34+k5₄₂ =v12+v32+z22+z32+z33+z42+z34

h02+h11+h03+h21+h13+h31+k5₄₃ =v13+v33+z23+z33+z34+z43

h11+h03+h21+h13+h32+h24+k5₄₄ =v14+v34+z31+z32+z24+z33+z43+z44

h01+h03+h12+h21+h04+h13h31+h23+k6₁₁ =v11+v31+v61+z21+z41+z42+z34+z43

h01+h21+h13+h22+h14+h32h24+k6₁₂ =v12+v12+v33+v62+z22+z31+z41+z33+z43

h01+h02+h21+h22+h14+h23h33+k6₁₃ =v13+v13+v34+v63+z23+z32+z42+z33+z43

h02+h11+h03+h12+h22+h24h34+k6₁₃ =v14+v14+v34+v64+z24+z33+z43+z33+z44

h01+h02+h11+h03+h12+h21h13+h14+k6₂₁ =v61+z31+z42+z34+z43+z44

h02+h03+h12+h04+h13+h22+h14+k6₂₂ =v62+z32+z33+z42+z43

h02+h11+h04+h14+h23+z31+k6₂₃ =v63+z41+z44

h01+h04+h13+h14+h24+z32+z41+k6₂₄ =v64+z34+z43+z44

h01+h02+h11+h12+h04+h23+k6₃₁ =v61+z41+z33+z42

h02+h04+h14+h24+k6₃₂ =v62+z31+z44

h01+h11+h03+h21+k6₃₃ =v63+z32+z41

h01+h11+h03+h22+h14+z32+z41+k6₃₄ =v64+z44

h01+h02+h11+h12+h04+h23+k6₄₁ =v01+v21+v41+v51+z11+z41+z33+z42

h02+h04+h14+h24+k6₄₂ =v02+v22+v42+v52+z12+z31+z44

h01+h11+h03+h21+k6₄₃ =v03+v23+v43+v53+z13+z32+z41

h01+h11+h03+h22+h14k6₄₄ =v04+v24+v44+v54+z14+z32+z41+z44

h02+h11+h03+h21+h13+k7₁₁ =v01+v21+v41+v51+z11+z71+z33+z34+z43

h11+h03+h12+h04+h22+h14+k7₁₂ =v02+v22+v42+v52+z12+z31+z41+z33+z42+z34

h11+h12+h04+h13+h23+k7₁₃ =v03+v23+v43+v53+z13+z32+v73+z42+z34+z43

h01+h02+h12+h14+h24+k7₁₄ =v04+v24+v44+v54+z14+z32+z33+z42+v74+z34

h01+h02+h11+h03+h04+k7₂₁ =v71+z31+z32+z41

h02+h03+h12+h04+k7₂₂ =v72+z32+z33+z42

h01+h04+h13+z31+z32+z41 +k7₂₃ =v73+z33+z34+z43

h01+h03+h04+h14+k7₂₄ =v74+z34   [formula 15]

[0097] Then, the next step S104 is executed to carry out amatricial-equation transformation process. In this process, vectors K,H, U and V are set as follows.

K=(k1₁₁, k1₁₂, . . . , k7₂₄)

H=(h01, h02, . . . , h44)

U=(z01, z02, . . . , z44)

V=(v01, v02, . . . , v74)   [formula 16]

[0098] With the vectors K, H, U and V set as expressed by the aboveequations, the simultaneous linear equation can be transformed into thefollowing matricial equation.

[0099] [Formula 17] ${M_{KH}\begin{pmatrix}{\,^{t}K} \\{\,^{t}H}\end{pmatrix}} = {M_{UV}\begin{pmatrix}{\,^{t}U} \\{\,^{t}V}\end{pmatrix}}$

[0100] It is to be noted that, in the above equation, symbols M_(KH) andM_(UV) each denote a GF(2) matrix comprising coefficients of thesimultaneous linear equation described above.

[0101] Then, the next step S105 is executed to carry out a unitarytransformation process.

[0102] Let symbol N_(r) denote the rank value of the matrix M_(UV) asfollows:

rank(M _(UV))=N _(r)   [formula 18]

[0103] Then, let symbol Ndenote the number of rows composing the matrixM_(UV). By multiplying both the left-hand and right-hand sides of thematricial equation by a row-deform unitary matrix Q from the left, thematrix M_(UV) can be deformed into a step matrix. In this process, asmall matrix consisting of (N_(m)-N_(r)) lowest rows of the matrixQM_(UV), becomes a null matrix.

[0104] Then, the next step S106 is executed to carry out a small-matrixselection process. Let symbol M*KH denote a small matrix consisting of(N_(m)-N_(r)) lowest rows of the matrix QM_(KH). In this case, the smallmatrix M*_(KH) becomes a null matrix (O) as expressed by the followingequation.

M*_(KH)=O   [formula 19]

[0105] Then, the next step S107 is executed to carry out alinear-relation equation generation process. This matricial equation istransformed into linear-relation equations, which are each associatedwith a row. Then, actual values are substituted for h01, h02, . . . ,and h44 to obtain the following relation equations:

0×c7=k1₁₁ +k1₂₁ +k1₂₂ +k1₂₄ +k1₃₁ +k1₃₂ +k1₃₄ +k1₄₂ +k1₄₄ +k2₁₂ +k2₁₄+k2₂₂ +k2₂₄ +k2₄₁

0×33=k1₁₂ +k1₂₁ +k1₂₂ +k1₂₃ +k1₃₁ +k1₃₂ +k1₃₃ +k1₄₁ +k1₄₃ +k2₁₁ +k2₁₃+k2₂₁ +k2₂₃ +k2₄₂

0×48=k1₁₃ +k1₂₂ +k1₂₄ +k1₃₂ +k1₃₄ +k1₄₁ +k1₄₂ +k1₄₄ +k2₁₁ +k2₁₂ +k2₁₄+k2₂₁ +k2₂₂ +k2₂₄ +k2₄₃

0×ef=k1₁₄ +k1₂₁ +k1₂₂ +k1₂₃ +k1₂₄ +k1₃₁ +k1₃₂ +k1₃₃ +k1₃₄ +k2₄₁ +k2₄₃+k1₄₄ +k2₁₁ +k2₁₃ +k2₁₄ +k2₂₁ +k2₂₃ +k2₂₄ +k2₄₄

0×c7=k1₂₁ +k1₂₁ +k1₃₁ +k2₁₁ +k3₄₁

0×33=k1₂₂ +k1₃₂ +k2₁₂ +k3₄₂

0×00=k1₂₃ +k1₃₃ +k1₄₁ +k2₁₂ +k2₁₃ +k2₂₁ +k3₄₁ +k3₄₂ +k3₄₃

0×dA=k1₂₄ +k1₃₄ +k1₄₃ +k2₁₁ +k2₁₂ +k2₁₃ +k2₂₃ +k2₂₃ +k2₄₂ +k4₁₁ +k4₂₁

0×7=k1₄₁ +k1₄₂ +k2₂₁ +k2₂₂ +k3₄₂ +k3₄₃ +k4₁₁ +k4₁₃ +k4₂₁ +k4₂₃

0×74=k1₄₂ +k1₄₃ +k2₁₁ +k2₁₂ +k2₂₂ +k2₂₃ +k3₄₂ +k3₄₃ +k4₁₁ +k4₁₂ +k4₂₁+k4₂₂

0×65=k1₄₃ +k2₁₂ +k2₁₄ +k2₂₃ +k3₄₂ +k4₁₃ +k4₁₄ +k4₂₃ +k4₂₄

0×33=k1₄₄ +k2₁₁ +k2₂₄ +k3₄₁ +k3₄₄

0×8α=k2₁₁ +k2₁₂ +k3₄₂ +k3₄₄ +k4₁₁ +k4₁₄ +k4₂₄ +k4₃₁

0×f7=k2₁₂ +k2₁₃ +k3₄₁ +k3₄₃ +k4₁₁ +k4₁₂ +k4₂₁ +k4₃₂

0×29=k2₁₃ +k2₁₄ +k3₄₁ +k3₄₂ +k3₄₄ +k4₁₁ +k4₁₂ +k4₁₃ +k4₂₁ +k4₂₂ +k4₃₃

0×α1=k2₁₄ +k3₄₁ +k3₄₄ +k4₁₁ +k4₂₂ +k4₂₃ +k4₂₄ +k4₃₁ +k4₃₂ +k4₃₃ +k4₃₄

0×41=k2₂₁ +k2₃₁ +k3₄₁ +k3₄₃ +k3₄₄ +k4₁₁ +k4₁₃ +k4₁₄ +k4₂₃ +k4₃₁ +k4₃₄

0×74=k2₂₂ +k2₃₂ +k3₄₁ +k3₄₂ +k3₄₃ +k4₁₁ +k4₁₂ +k4₁₃ +k4₂₃ +k4₂₄ +k4₃₁+k4₃₂ +k4₃₄

0×f4=k2₂₃ +k2₃₃ +k3₄₁ +k3₄₂ +k3₄₃ +k3₄₄ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₄ +k4₂₄+k4₃₁ +k4₃₂ +k4₃₃

0×57=k2₂₄ +k2₃₄ +k4₂₄ +k4₃₄

0×f6=k2₄₁ +k3₁₁ +k3₂₁ +k3₄₁ +k3₄₂ +k3₄₃ +k4₁₁ +k4₁₂ +k4₁₃ +k4₂₁ +k4₂₃+k4₂₄ +k4₃₂ +k4₃₄

0×7c=k2₄₂ +k3₁₂ +k3₂₂ +k3₄₂ +k4₁₂ +k4₂₂ +k4₂₃ +k4₂₄ +k4₃₃ +k4₃₄

0×43=k2₄₃ +k3₁₃ +k3₂₃ +k3₄₁ +k3₄₂ +k3₄₃ +k4₁₁ +k4₁₂ +k4₂₁ +k4₃₂ k4₃₃

0×5f=k2₄₄ +k3₁₄ +k3₂₄ +k3₄₃ +k4₁₃ +k4₂₂ +k4₂₄ +k4₃₂ +k4₃₃ +k4₃₄

0×7d=k3₁₁ +k3₄₁ +k3₄₂ +k3₄₃ +k3₄₄ +k4₁₁ +k12+4 ₁₃ +k4₁₄ +k4₂₁ +k4₂₄ k4₃₂+k4₃₃ +k5₄₁

0×2d=k3₁₂ +k3₄₁ +k3₄₂ +k4₁₁ +k4₁₂ +k4₂₂ +k4₂₃ +k4₃₁ +k4₃₃ +k4₄₂

0×02=k3₁₃ +k3₄₄ +k4₁₄ +k4₂₁ +k4₂₃ +k4₃₁ +k4₃₃ +k4₃₄ +k4₄₃

0×de=k3₁₄ +k3₄₂ +k3₄₃ +k3₄₄ +k4₁₂ +k4₁₃ +k4₁₄ +k4₂₂ +k4₃₃ +k4₃₄ +k5₄₄

0×8α=k3₂₁ +k3₃₁ +k3₄₂ +k3₄₃ +k3₄₄ +k4₁₂ +k4₁₃ +k4₁₄ +k4₂₄ +k4₃₂ +k4₃₃

0×7f=k3₂₂ +k3₃₂ +k3₄₁ +k3₄₂ +k3₄₃ +k3₄₄ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₄ +k4₂₃+k4₂₄ +k4₃₁ +k4₃₂

0×88=k3₂₃ +k3₃₃ +k3₄₁ +k3₄₂ +k4₁₁ +k4₁₂ +k4₂₁ +k4₂₃ +k4₂₄ +k4₃₂ +k4₃₃+k4₃₄

0×54=k3₂₄ +k3₃₄ +k3₄₂ +k3₄₃ +k4₁₂ +k4₁₃ +k4₂₂ +k4₂₄ +k4₃₃ +k4₃₄

0×7f=k3₄₁ +k3₄₂ +k4₁₂ +k4₂₁ +k4₂₃ +k4₂₄ +k4₃₂ +k4₃₃ +k4₃₄ +k5₁₁ +k5₂₁

0×7f=k3₄₂ +k4₁₁ +k4₁₂ +k4₁₃ +k4₂₁ +k4₂₄ +k4₃₁ +k4₃₂ +k4₃₄ +k5₁₁ +k5₁₃+k5₂₁ +k5₂₃

0×8α=k3₄₃ +k3₄₄ +k4₁₁ +k4₁₃ +k4₂₁ +k4₃₁ +k4₃₃ +k4₃₄ +k5₁₁ +k5₁₄ +k5₂₁+k5₂₄

0×00=k3₄₄ +k4₁₂ +k4₁₄ +k4₂₂ +k4₃₂ +k4₃₄ +k5₁₂ +k5₂₂

0×f7=k4₁₁ +k4₁₃ +k4₂₃ +k4₃₃ +k4₄₁ +k5₁₁ +k5₁₃ +k5₂₁ +k5₂₃ +k5₄₁

0×29=k4₁₂ +k4₁₃ +k4₁₄ +k4₂₁ +k4₂₃ +k4₂₄ +k4₃₁ +k4₃₃ +k4₃₄ +k4₄₁ +k4₄₂+k5₁₂ +k5₁₃ +k5₁₄ +k5₂₂ +k5₂₃ +k5₂₄ +k5₄₁ +k5₄₂

0×2b =k4₁₃ +k4₁₄ +k4₂₂ +k4₂₄ +k4₃₂ +k4₃₄ +k4₄₂ +k4₄₃ +k5₁₃ +k5₁₄ +k5₂₃+k5₂₄ +k5₄₂ +k5₄₃

0×88=k4₁₄ +k4₂₁ +k4₂₃ +k4₃₁ +k4₃₃ +k4₄₁ +k4₄₃ +k4₄₄ +k5₁₄ +k5₂₄ +k5₄₁+k5₄₃ +k5₄₄

0×43=k4₂₁ +k4₃₁ +k5₄₁ +k6₁₁ +k6₂₁

0×e0=k4₂₂ +k4₂₄ +k4₃₂ +k4₃₄ +k4₄₁ +k4₄₂ +k4₄₄ +k5₄₄ +k6₁₁ +k6₁₂ +k6₂₁+k6₃₂

0×eb=k4₂₃ +k4₂₄ +k4₃₃ +k4₃₄ +k4₄₁ +k5₄₃ +k6₁₁ +k6₁₃ +k6₂₁ +k6₃₃

0×81=k4₂₄ +k4₃₄ +k4₄₂ +k4₄₃ +k5₄₃ +k6₁₂ +k6₂₂

0×7e=k4₄₁ +k5₄₁ +k5₄₃ +k6₁₃ +k6₂₃

0×dd=k4₄₂ +k4₄₃ +k4₄₄ +k5₄₂ +k5₄₃ +k6₁₄ +k6₂₄

0×00=k4₄₃ +k4₄₄ +k5₄₃ +k6₁₄ +k6₃₄

0×00=k4₄₄ +k5₄₁ +k5₄₄ +k6₁₁ +k6₃₁

0×f7=k5₅₁ +k5₄₁ +k6₁₁ +k6₃₁ +k6₄₁

0×14=k5₁₂ +k5₄₁ +k5₄₃ +k5₄₄ +k6₁₁ +k6₁₃ +k6₁₄ +k6₂₁ +k6₂₃ +k6₃₄ +k6₄₂

0×23=k5₁₃ +k5₄₁ +k5₄₂ +k6₁₁ +k6₁₂ +k6₂₂ +k6₂₄ +k6₃₁ +k6₃₄ +k6₄₃

0×8α=k5₁₄ +k5₄₄ +k6₁₄ +k6₃₄ +k6₄₄

0×v4=k5₂₁ +k5₃₁ +k5₄₁ +k6₄₂ +k6₁₁ +k6₁₂ +k6₂₁ +k6₃₂

0×0b=k5₂₂ +k5₃₂ +k5₄₂ +k6₄₃ +k6₁₂ +k6₁₃ +k6₂₂ +k6₃₃

0×00=k5₂₃ +k5₃₃ +k5₄₁ +k5₄₂ +k5₄₄ +k6₁₁ +k6₁₂ +k6₁₄ +k6₃₁ +k6₃₂ +k6₃₄

0×00=k5₂₄ +k5₃₄ +k5₄₁ +k5₄₃ +k5₄₄ +k6₁₁ +k6₁₃ +k6₁₄ +k6₃₁ +k6₃₃ +k6₃₄

0×c7=k5₄₁ +k5₄₂ +k5₄₃ +k5₄₄ +k6₁₁ +k6₁₂ +k6₁₃ +k6₁₄ +k6₂₁ +k6₂₃ +k6₃₂+k6₃₄ +k6₄₁ +k7₁₁ +k7₂₁

0×fc=k5₄₂ +k6₁₂ +k6₂₁ +k6₃₁ +k6₃₂ +k6₄₃ +k7₁₃ +k7₂₃

0×18=k5₄₃ +k5₄₄ +k6₁₃ +k6₁₄ +k6₂₁ +k6₂₂ +k6₃₁ +k6₃₂ +k6₃₃ +k6₃₄ +k6₄₃+k6₄₄ +k7₁₃ +k7₁₄ +k7₂₃ +k7₂₄

0×f4=k6₂₁ +k6₂₂ +k6₂₃ +k6₂₄ +k6₃₁ +k6₃₂ +k6₃₃ +k6₃₄ +k6₄₁ +k6₄₂ +k7₁₁+k7₁₂ +k7₂₁ +k7₂₂ [formula 20]

[0106] Here, the following equation holds true.

rank(M*K)=N _(m)-N _(r)   [formula 21]

[0107] Thus, the above 60 linear-relation equations are linear-relationequations independent of each other. It is therefore obvious that(2⁶⁰-1) linear-relation equations obtained from linear concatenation ofany of the 60 equations on the GF(2) hold true. If the number of suchlinear-relation equations is large, it is feared that a new attack thatthe designer of the encryption method is not aware of is brought about.For this reason, the total number of linear-relation equations obtainedby adoption of the method described above can be used as an indicatorfor the evaluation of the encryption level.

[0108] [Second Embodiment of Encryption Level Indicator CalculationProcess]

[0109] As a second embodiment of the encryption level indicatorcalculation process provided by the present invention, a typical processof applying an encryption level evaluation method provided by thepresent invention to ‘Hierocrypt-3’ is explained in detail.‘Hierocrypt-3’ is the name of an AES-compatible block encryption processproposed by Toshiba. The ‘Hierocrypt-3’ block encryption process is acommon-key block encryption process with a block length of 128 bits anda key length of 128, 192 or 256 bits. A typical encryption processexplained below is a process with a key length of 256 bits.

[0110] First of all, the step S101 of the flowchart shown in FIG. 1 isexplained. As described earlier, at this step, an encryption processingalgorithm is set. This step is executed as a process to set the‘Hierocrypt-3’ block encryption algorithm proposed by Toshiba.

[0111] First of all, a matrix P32 is defined as follows:

[0112] [Formula 22] ${P32} = \begin{pmatrix}{I4} & {O4} & {I4} & {O4} \\{O4} & {I4} & {O4} & {I4} \\{O4} & {I4} & {I4} & {I4} \\{I4} & {O4} & {I4} & {I4}\end{pmatrix}$

[0113] Let symbol P32I denote the inverse matrix of the matrix P32.Next, matrices M51, M52, MB1 and MB2 are defined as follows:

[0114] [Formula 231 $\begin{matrix}{{M51} = \begin{pmatrix}1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\0 & 1 & 0 & 1\end{pmatrix}} \\{{M52} = \begin{pmatrix}1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 \\0 & 0 & 1 & 1 \\1 & 1 & 1 & 0\end{pmatrix}} \\{{M\quad {B1}} = \begin{pmatrix}0 & 1 & 0 & 1 \\1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 0 & 1 & 1\end{pmatrix}} \\{{M\quad {B2}} = \begin{pmatrix}1 & 1 & 0 & 0 \\0 & 1 & 1 & 0 \\1 & 0 & 1 & 1 \\1 & 0 & 0 & 1\end{pmatrix}}\end{matrix}$

[0115] Then, matrices M51, M52, MB1 and MB2 are defined, being expressedin terms of the matrices M5 and MB as follows:

[0116] [Formula 24] $\begin{matrix}{{M5} = \begin{pmatrix}{M51} & {O4} & {O4} & {O4} \\{O4} & {M52} & {O4} & {O4} \\{O4} & {O4} & {M51} & {O4} \\{O4} & {O4} & {O4} & {M52}\end{pmatrix}} \\{{MB} = \begin{pmatrix}{MB1} & {O4} & {O4} & {O4} \\{O4} & {MB2} & {O4} & {O4} \\{O4} & {O4} & {MB1} & {O4} \\{O4} & {O4} & {O4} & {MB2}\end{pmatrix}}\end{matrix}$

[0117] Next, round dependent constant vectors Gi (where i=0, . . . 9)are defined as follows:

[0118] [Formula 25]

G0=(h11,h12,h13,h14,h01,h02,h03,h04,0,0,0,0,0,0,0,0)

G1=(h21,h22,h23,h24,h31,h32,h33,h34,0,0,0,0,0,0,0,0)

G2=(h31,h32,h33,h34,h01,h02,h03,h04,0,0,0,0,0,0,0,0)

G3=(h11,h12,h13,h14,h31,h32,h33,h34,0,0,0,0,0,0,0,0)

G4=(h21,h22,h23,h24,h11,h12,h13,h14,0,0,0,0,0,0,0,0)

G5=(h01,h02,h03,h04,h21,h22,h23,h24,0,0,0,0,0,0,0,0)

G6=(h01,h02,h03,h04,h21,h22,h23,h24,0,0,0,0,0,0,0,0)

G7=(h21,h22,h23,h24,h11,h12,h13,h14,0,0,0,0,0,0,0,0)

G8=(h11,h12,h13,h14,h31,h32,h33,h34,0,0,0,0,0,0,0,0)

G9=(h31,h32,h33,h34,h01,h02,h03,h04,0,0,0,0,0,0,0,0)

[0119] It is to be noted that a vector HH with constants in the aboveequations used as elements is the same as the vector HH of the firstembodiment implementing the encryption level indicator calculationprocess as described earlier.

[0120] Next, a vector ZZ with its elements composing the right half of asequence of initial values of the key-scheduling part is defined asfollows.

ZZ=(z31, z32, z33, z34, z35, z36, z37, z38, z41, z42, z43; z44, z45,z46, z47, z48)   [formula 26]

[0121] By using these, the right half of the key-scheduling part in theHierocrypt-3 common-key encryption algorithm is expressed below. It isto be noted that the operator + used in the following expressions is anadditive operator in the Galois field GF(2).

Z0M5*ZZ+G0

W0=P32*Z0

Z1=M5*W0+G1

W1=P32*Z1

Z2=M5*W1+G2

W2=P32*Z2

Z3=M5*W2+G3

W3=P32*Z3

Z4=M5*W3+G4

W4=P32*Z4

Z5=M5*W4+G5

W6=MB*(Z5+G6)

Z6=P32I*W6

W7=MB*(Z6+G7)

Z7=P32I*W7

W8=MB*(Z7+G8)

Z8=P32I*W8

W9=MB*(Z8+G9)

Z9=P32I*W9   [formula 27]

[0122] Symbols Z0, Z1, Z2, Z3, Z4, Z5, Z6, Z7, Z8, Z9, W0, W1, W2, W3,W5, W6, W7, W8 and W9 used in the above equations form the right half ofthe sequence of intermediate values of the key-scheduling part.

[0123] Next, these intermediate values are expressed by being split inaccordance with the following equations.

Zn=Zn ₃ ||ZN ₄

Wn=Wn ₁ ||WN ₃   [formula 28]

[0124] Symbol || used in the above equations denotes a concatenation ofvectors.

[0125] Next, let non-linear transformation part outputs of rounds be V0,V1, V2, V3, V4, V5, V6, V7, V8 and V9. Each of the outputs is a vectorconsisting of eight elements as follows.

V0=(v01, v02, v03, v04, v05, v06, v07, v08)

V1=(v11, v12, v13, v14, v15, v16, v17, v18)

V2=(v21, v22, v23, v24, v25, v26, v27, v28)

V3=(v31, v32, v33, v34, v35, v36, v37, v38)

V4=(v41, v42, v43, v44, v45, v46, v47, v48)

V5=(v51, v52, v53, v54, v55, v56, v57, v58)

V6=(v61, v62, v63, v64, v65, v66, v67, v68)

V7=(v71, v72, v73, v74, v75, v76, v77, V78)

V8=(v81, v82, v83, v84, v85, v86, v87, v88)

V9=(v91, v92, v93, v94, v95, v96, v97, v98)   [formula 29]

[0126] Here, vectors Z₁ and Z₂ are set as follows.

Z₁=(z11, z12, z13, z14, z15, z16, z17, z18)

Z₂=(z21, z22, z23, z24, z25, z26, z27, z28)   [formula 30]

[0127] With the vectors Z₁ and Z₂ set as described above, the left halfof the sequence of the key-scheduling part in the Hierocrypt-3common-key encryption algorithm is expressed as follows.

X0₁=Z₂

Z0₂ =Z ₁ +V0

Z1₁=Z0₂

Z1=Z0₁ +V1

Z2₁=Z1₁

Z2₂ =Z1₃ +V2

Z3₁=Z2₂

Z3₂ =Z2₁ +V3

Z4₁=Z3₂

Z4₂ =Z3₁ +V4

Z5₁=Z₄ ₂

Z5₂ =Z4₂ +V5

Z6₂ =Z5₂ +V6

Z6₂=Z5₂

Z7₁ =Z6₂ +V7

Z7₂=Z6₃

Z8₁ =Z7₂ +V8

Z8₂=Z7₁

Z9₁ =Z8₂ +V9

Z9₂=Z8₁   [formula 31]

[0128] Symbols Z0₁, Z0₂, Z1₁, Z1₂, Z2₁, Z2₂, Z3₁, Z3₂, Z4₁, Z4₂, Z5₁,Z5₂, Z6₁, Z6₂, Z7₁, Z7₂, Z8₁, Z8₂, Z9₁ and Z9₂ used in the aboveequations form the left half of the sequence of intermediate values ofthe key-scheduling part. By using the intermediate values obtained asdescribed above, round keys K1₁, K1₂, K1₃, K1₄, K2₁, . . . , K9₁ and K9₂are expressed as follows:

K1₁ =Z0₂ +V1

K1₂ =Z1₃ +V1

K1₃ =Z1₄ +V1

K1₄ =Z0₃ +Z1₄

K2₁ =Z1₂ +V2

K2₂ =Z2₃ +V2

K2₃ =Z2₄ +V2

K2₄ =Z1₂ +Z2₄

K3₁ =Z2₁ +V3

K3₂ =Z3₃ +V3

K3₃ =Z3₄ +V3

K3₄ =Z2₂ +Z3₄

K4₁ =Z3₁ +V4

K4₂ =Z4₃ +V4

K4₃ =Z4₄ +V4

K4₄ =Z3₂ +Z4₄

K5₁ =Z4₁ +V5

K5₂ =Z5₃ +V5

K5₃ =Z5₄ +V5

K5₄ =Z4₂ +Z5₄

K6₁ =Z6₁ +Z5₃

K6₂ =W6₁ +V6

K6₃ =W6₂ +V6

K6₄ =Z5₁ +W6₂

K7₁ =Z7₁ +Z6₃

K7₂ =W7₁ +V7

K7₃ =W7₂ +V7

K7₄ =Z6₁ +W7₂

K8₁ =Z8₁ +Z7₃

K8₃ =W8₁ +V8

K8₃ =Z7₁ +W8₂

K9₁ =Z9₁ +Z8₃

K9₂ =W9₁ +V9

K9₃ =W9₂ +V9

K9₄ =Z8₁+W9₃   [formula 32]

[0129] It is to be noted that symbols K11, K12, K13, K14, K21, . . , K91and K92 each denote a vector consisting of eight elements.

[0130] The following description explains the step S102 of carrying outa process to eliminate intermediate variables in the processingrepresented by the flowchart shown in FIG. 1. If the eight elements ofeach of the vectors K11, K12, K13, K14, K21, . . . , K91 and K92 areexpressed by their actual values, the vectors K11, K12, K13, K14, K21, .. . , K91 and K92 can be expressed as follows:

[0131] [Formula 33] $\begin{matrix}{{K1}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v11} + {z21}} \\{{v12} + {z22}}\end{matrix} \\{{v13} + {z23}}\end{matrix} \\{{v14} + {z24}}\end{matrix} \\{{v15} + {z25}}\end{matrix} \\{{v16} + {z26}}\end{matrix} \\{{v17} + {z27}}\end{matrix} \\{{v18} + {z28}}\end{pmatrix}} \\{{K1}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h11} + {h21} + {h13} + {v11} + {z32} + {z42}} \\{{h11} + {h12} + {h22} + {h14} + {v12} + {z33} + \quad {z43}}\end{matrix} \\{{h11} + {h12} + {h13} + {h23} + {v13} + {z31} + {z41} + {z34} + {z44}}\end{matrix} \\{{h12} + {h14} + {h24} + {v14} + {z31} + \quad {z41}}\end{matrix} \\{{h01} + {h02} + {h03} + {h04} + {h31} + {v15} + \quad {z36} + {z46} + {z38} + {z48}}\end{matrix} \\{{h02} + \quad {h03} + {h04} + {h32} + {v16} + \quad {z35} + \quad {z45} + {z37} + {z47}}\end{matrix} \\{\quad {{h03} + {h04} + {h33} + {v17} + {z35} + {z36} + {z45} + {z46} + {z38} + {z48}}}\end{matrix} \\{{h01} + {h02} + {h03} + \quad {h34} + {v18} + {z35} + {z45} + {z37} + {z38} +} \\{{z47} + {z48}}\end{pmatrix}} \\{{K1}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h03} + {v11} + {z42} + {z35} + {z36} + {z45} + {z46}} \\{{h01} + {h02} + {h04} + {v12} + {z43} + {z36} + {z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h03} + {v13} + \quad {z41} + {z35} + \quad {z44} + {z45} + {z37} + {z38} +} \\{{z47} + \quad {z48}}\end{matrix} \\{{h02} + {h04} + {v14} + {z41} + {z35} + {z45} + {z38} + {z48}}\end{matrix} \\{{h11} + {h12} + {h13} + {h14} + {v15} + {z31} + \quad {z32} + {z41} + {z42} + {z46} +} \\{\quad {z48}}\end{matrix} \\{{h12} + {h13} + {h14} + {v16} + {z32} + {z33} + {z42} + {z43} + {z45} + \quad {z47}}\end{matrix} \\{{h13} + {h14} + {v17} + {z31} + {z41} + {z33} + {z34} + {z43} + \quad {z44} + {z45} +} \\{{z46} + {z48}}\end{matrix} \\{{h11} + {h12} + {h13} + {z31} + {v18} + {z41} + {z34} + {z44} + {z45} +} \\{{z47} + {z48}}\end{pmatrix}} \\{{K1}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h\quad 03} + {v01} + {z11} + {z42} + {z35} + \quad {z36} + \quad {z45} + {z46}} \\{{h01} + {h02} + {h04} + {v02} + {z12} + {z43} + {z36} + {z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h03} + {v03} + {z13} + {z41} + {z35} + {z44} + {z45} + {z37} +} \\{{z38} + {z47} + \quad {z48}}\end{matrix} \\{{h02} + {h04} + {v04} + {z14} + {z41} + {z35} + \quad {z45} + {z38} + {z48}}\end{matrix} \\{{h11} + {h12} + {{h1}\quad 3} + {h14} + {v05} + {z31} + {z32} + {z41} + {z15} +} \\{{z42} + {z46} + {z48}}\end{matrix} \\{\quad {{h12} + {h13} + {h14} + {v06} + {z32} + \quad {z33} + {z42} + {z16} + {z43} +}} \\{{z45} + {z47}}\end{matrix} \\{{h13} + {h14} + {v07} + {z31} + \quad {z41} + {z33} + {z34} + {z43} + {z17} + {z44} +} \\{{z45} + \quad {z46} + {z48}}\end{matrix} \\{{h11} + {h12} + {h13} + {v08} + {z31} + {z41} + {z34} + {z44} + {z18} +} \\{{z45} + {z47} + \quad {z48}}\end{pmatrix}} \\{{K2}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v01} + {v21} + {z21}} \\{{v02} + {v22} + {z22}}\end{matrix} \\{{v03} + {v23} + {z23}}\end{matrix} \\{{v04} + {v24} + {z24}}\end{matrix} \\{{v05} + {v25} + {z25}}\end{matrix} \\{{v06} + {v26} + {z26}}\end{matrix} \\{{v07} + {v27} + {z27}}\end{matrix} \\{{v08} + {v28} + {z28}}\end{pmatrix}} \\{{K2}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h12} + {h21} + {h31} + {h\quad 23} + {\tau 21} + \quad {z31} + {z32} + {z34} +} \\{\quad {{z36} + \quad {z37} + {z45} + {z38} + {z47} + {z48}}} \\{{h\quad 08}\quad + {h21} + {h13} + {h22} + {h32} + {h24} + {v22} + {z31} + {z32} +} \\{{z33} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h11} + {h21} + {h04} + {h22} + {h23} + {h33} + {v23} + {z31} +} \\{{z32} + {z33} + {z34} + {z38} + {z48}}\end{matrix} \\{{h01} + {h11} + {h22} + {h24} + {h34} + {v24} + {z31} + {z33} + {z35} +} \\{{z36} + {z45} + {z37} + {z46} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h12} + {h04} + {h31} + {h14} + {h32} + {h33} + {h34} +} \\{{v25} + {z31} + {z41} + {z35} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h13} + {h32} + {h33} + {h34} + {v26} +} \\{{z32} + {z42} + {z35} + {z36}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h04} + {h14} + {h33} + {h34} +} \\{{v27} + {z33} + {z43} + {z36} + {z37}}\end{matrix} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h32} + {h33} + {v28} +} \\{{z34} + {z44} + {z37}}\end{pmatrix}} \\{{K2}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h12} + {h31} + {h33} + {v21} + {z32} + {z41} + {z33} +} \\{{z34} + {z43} + {z35} + {z36} + {z37} + {z46} + {z38} + {z47} + {z48}} \\{{h02} + {h12} + {h13} + {h31} + {h32} + {h34} + {v22} + {z41} + {z33} +} \\{{z42} + {z34} + {z44} + {z36} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h11} + {h03} + {h13} + {h31} + {h14} + {h32} + {h33} + {v23} + {z41} +} \\{{z42} + {z34} + {z43} + {z37} + {z38} + {z48}}\end{matrix} \\{{h11} + {h04} + {h14} + {h32} + {h34} + {v24} + {z31} + {z32} + {z33} +} \\{{z42} + {z34} + {z35} + {z44} + {z36} + {z45} + {z37} + {z46} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h21} + {h22} + {h14} + {h23} + {h24} +} \\{{v25} + {z31} + {z32} + {z41} + {z33} + {z34} + {z35} + {z48}}\end{matrix} \\{{h02} + {h11} + {h03} + {h12} + {h22} + {h23} + {h24} + {v26} + {z32} +} \\{{z33} + {z42} + {z34} + {z36} + {z45}}\end{matrix} \\{{h01} + {h03} + {h12} + {h04} + {h13} + {h23} + {h24} + {v27} +} \\{{z33} + {z34} + {z43} + {z37} + {z46}}\end{matrix} \\{{h01} + {h21} + {h04} + {h13} + {h22} + {h23} + {z31} + {z32} +} \\{{v28} + {z33} + {z44} + {z38} + {z47} + {z48}}\end{pmatrix}} \\{{K2}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h12} + {h31} + {h33} + {v11} + {z21} + {z32} + {z41} +} \\{{z33} + {z34} + {z43} + {z35} + {z36} + {z37} + {z46} + {z38} + {z47} + {z48}} \\{{h02} + {h12} + {h13} + {h31} + {h32} + {h34} + {v12} + {z22} + {z41} +} \\{{z33} + {z42} + {z34} + {z44} + {z36} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h11} + {h03} + {h13} + {h31} + {h14} + {h32} + {h33} + {v13} + {z23} +} \\{{z41} + {z42} + {z34} + {z43} + {z37} + {z38} + {z48}}\end{matrix} \\{{h11} + {h04} + {h14} + {h32} + {h34} + {v14} + {z31} + {z32} + {z24} + {z34} +} \\{{z35} + {z44} + {z36} + {z45} + {z37} + {z46} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h21} + {h22} + {h14} + {h23} + {h24} + {v25} +} \\{{z31} + {z32} + {z41} + {z33} + {z34} + {z35} + {z48}}\end{matrix} \\{{h02} + {h11} + {h03} + {h12} + {h22} + {h23} + {h24} + {v16} + {z32} +} \\{{z33} + {z42} + {z34} + {z26} + {z36} + {z45}}\end{matrix} \\{{h01} + {h03} + {h12} + {h04} + {h13} + {h23} + {h24} + {v17} + {z33} +} \\{{z34} + {z43} + {z37} + {z46}}\end{matrix} \\{{h01} + {h21} + {h04} + {h13} + {h22} + {h23} + {z31} + {v18} + {z32}} \\{{z33} + {z44} + {z28} + {z38} + {z47} + {z48}}\end{pmatrix}} \\{{K3}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v11} + {v31} + {z21}} \\{{v12} + {v32} + {z22}}\end{matrix} \\{{v13} + {v33} + {z23}}\end{matrix} \\{{v14} + {v34} + {z24}}\end{matrix} \\{{v15} + {v35} + {z25}}\end{matrix} \\{{v16} + {v36} + {z26}}\end{matrix} \\{{v17} + {v37} + {z27}}\end{matrix} \\{{v18} + {v38} + {z28}}\end{pmatrix}} \\{{K3}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h03} + {h04} + {h13} + {h22} + {h31} + {h32} + {h33} + {v31} +} \\{{z32} + {z42} + {z35} + {z37}} \\{{h11} + {h03} + {h04} + {h31} + {h14} + {h23} + {h32} + {h33} + {h34} +} \\{{v32} + {z33} + {z43} + {z35} + {z36} + {z38}}\end{matrix} \\{{h11} + {h12} + {h21} + {h04} + {h32} + {h24} + {h33} + {h34} + {v33} +} \\{{z31} + {z41} + {z34} + {z35} + {z44} + {z36} + {z37}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h04} + {h31} + {h32} + {h34} +} \\{{v34} + {z31} + {z41} + {z36} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h22} + {h31} + {h32} + {h24} + {h34} +} \\{{v35} + {z31} + {z33} + {z35} + {z36} + {z45} + {z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h04} + {h31} + {h23} + {h32} +} \\{{h33} + {z31} + {v36} + {z32} + {z34} + {z35} + {z36} + {z45} + {z37} +} \\{{z46} + {z38} + {z47} + {z48}}\end{matrix} \\{{h02} + {h03} + {h21} + {h04} + {h13} + {h22} + {h31} + {h32} + {h24} + {h33} +} \\{{h34} + {z31} + {z32} + {v37} + {z33} + {z38} + {z37} + {z46} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h21} + {h04} + {h31} + {h14} + {h23} + {h24} + {h33} + {z32} +} \\{{v38} + {z34} + {z35} + {z36} + {z45} + {z46} + {z38} + {z48}}\end{pmatrix}} \\{{K3}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h03} + {h21} + {h04} + {h22} + {h31} + {v31} + {z42} + {z35} +} \\{{z38} + {z47} + {z48}} \\{{h02} + {h03} + {h04} + {h22} + {h23} + {h32} + {\tau 32} + {z43} + {z35} +} \\{{z36} + {z48}}\end{matrix} \\{{h03} + {h21} + {h04} + {h23} + {h24} + {h33} + {\tau 33} + {z41} + {z44} +} \\{{z36} + {z45} + {z37}}\end{matrix} \\{{h01} + {h02} + {h03} + {h21} + {h24} + {h34} + {v34} + {z41} + {z37} +} \\{{z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h21} + {h13} + {h24} + {h33} + {h34} + {v35} + {z31} +} \\{{z34} + {z43} + {z44} + {z45} + {z46} + {z47}}\end{matrix} \\{{h11} + {h12} + {h21} + {h13} + {h22} + {h14} + {h34} + {z31} + {v36} +} \\{{z32} + {z44} + {z45} + {z46} + {z47} + {z48}}\end{matrix} \\{{h12} + {h13} + {h22} + {h31} + {h14} + {h23} + {z32} + {z41} + {v37} +} \\{{z33} + {z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h14} + {h23} + {h32} + {h33} + {h34} + {z33} + {z42} +} \\{{v38} + {z43} + {z44} + {z45} + {z46} + {z48}}\end{pmatrix}} \\{{K3}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h03} + {h21} + {h04} + {h22} + {h31} + {v01} + {v21} +} \\{{z11} + {z42} + {z35} + {z38} + {z47} + {z48}} \\{{h02} + {h03} + {h04} + {h22} + {h23} + {h32} + {v02} + {v22} + {z12} +} \\{{z43} + {z35} + {z36} + {z48}}\end{matrix} \\{{h03} + {h21} + {h04} + {h23} + {h24} + {h33} + {v03} + {v23} + {z13} +} \\{{z41} + {z44} + {z36} + {z45} + {z37}}\end{matrix} \\{{h01} + {h02} + {h03} + {h21} + {h24} + {h34} + {v04} + {v24} + {z14} +} \\{{z41} + {z37} + {z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h21} + {h13} + {h24} + {h33} + {h34} + {v05} + {v25} +} \\{{z31} + {z15} + {z34} + {z43} + {z44} + {z45} + {z46} + {z47}}\end{matrix} \\{{h11} + {h12} + {h21} + {h13} + {h22} + {h14} + {h34} + {v06} + {v26} +} \\{{z31} + {z32} + {z16} + {z44} + {z45} + {z46} + {z47} + {z48}}\end{matrix} \\{{h12} + {h13} + {h22} + {h31} + {h14} + {h23} + {v07} + {v27} + {z32} +} \\{{z41} + {z33} + {z17} + {z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h14} + {h23} + {h32} + {h33} + {h34} + {v08} + {v28} +} \\{{z33} + {z42} + {z43} + {z44} + {z18} + {z45} + {z46} + {z48}}\end{pmatrix}} \\{{K4}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v01} + {v21} + {v41} + {z11}} \\{{v02} + {v22} + {v42} + {z12}}\end{matrix} \\{{v03} + {v23} + {v43} + {z13}}\end{matrix} \\{{v04} + {v24} + {v44} + {z14}}\end{matrix} \\{{v05} + {v25} + {v45} + {z15}}\end{matrix} \\{{v06} + {v26} + {v46} + {z16}}\end{matrix} \\{{v07} + {v27} + {v47} + {z17}}\end{matrix} \\{{v08} + {v28} + {v48} + {z18}}\end{pmatrix}} \\{{K4}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {h12} + {h13} + {h23} + {h33} + {h34} + {v41} +} \\{{z31} + {z32} + {z34} + {z35} + {z45} + {z37} + {z38} + {z47} + {z48}} \\{{h01} + {h02} + {h11} + {h12} + {h21} + {h04} + {h13} + {h14} + {h24} +} \\{{h34} + {v42} + {z31} + {z32} + {z33} + {z36} + {z46} + {z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h13} + {h22} + {h31} + {h14} +} \\{{v43} + {z31} + {z32} + {z33} + {z34} + {z35} + {z45} + {z37} + {z47}}\end{matrix} \\{{h02} + {h11} + {h12} + {h04} + {h22} + {h14} + {h32} + {h33} + {h34} +} \\{{v44} + {z31} + {z33} + {z36} + {z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h11} + {h03} + {h12} + {h23} + {h04} + {h31} + {h33} + {h34} +} \\{{z31} + {v45} + {z32} + {z41} + {z42} + {z34} + {z35} + {z44} + {z37} + {z38}}\end{matrix} \\{{h02} + {h12} + {h04} + {h13} + {h22} + {h32} + {h34} + {z31} + {z32} + {z41} +} \\{{v46} + {z33} + {z42} + {z43} + {z36} + {z38}}\end{matrix} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h23} + {h33} + {z31} +} \\{{z32} + {z41} + {z33} + {z42} + {v47} + {z34} + {z43} + {z35} + {z44} + {z37}}\end{matrix} \\{{h02} + {h11} + {h03} + {h14} + {h32} + {h24} + {h33} + {z31} + {z41} +} \\{{z33} + {z43} + {v48} + {z36} + {z37}}\end{pmatrix}} \\{{K4}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h04} + {h13} + {h14} + {h33} + {h34} + {v41} + {z31} + {z33} +} \\{{z42} + {z34} + {z43} + {z36} + {z45} + {z37} + {z47} + {z48}} \\{{h01} + {h03} + {h14} + {h34} + {v42} + {z32} + {z41} + {z34} + {z43} +} \\{{z35} + {z44} + {z37} + {z46} + {z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h04} + {h31} + {v43} + {z31} + {z33} + {z42} +} \\{{z44} + {z36} + {z45} + {z38} + {z47}}\end{matrix} \\{{h01} + {h03} + {h12} + {h04} + {h13} + {h14} + {h32} + {h33} + {h34} +} \\{{v44} + {z32} + {z41} + {z33} + {z42} + {z35} + {z36} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h21} + {h04} + {h22} + {h31} + {h23} +} \\{{h34} + {v45} + {z32} + {z41} + {z33} + {z42} + {z35} + {z44} +} \\{{z36} + {z46} + {z38} + {z47}}\end{matrix} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h22} + {h31} + {h23} +} \\{{h32} + {h24} + {z31} + {z41} + {v46} + {z33} + {z42} + {z34} + {z43} +} \\{{z35} + {z36} + {z45} + {z37} + {z47} + {z48}}\end{matrix} \\{{h03} + {h04} + {h13} + {h22} + {h23} + {h32} + {h24} + {h33} + {z32} +} \\{{z41} + {z42} + {v47} + {z34} + {z43} + {z35} + {z44} + {z36} + {z37} +} \\{{z46} + {z38} + {z48}} \\{{h01} + {h02} + {h03} + {h21} + {h22} + {h14} + {h24} + {h33} + {z31} +} \\{{z32} + {z41} + {z43} + {v48} + {z35} + {z45} + {z37} + {z46}}\end{pmatrix}} \\{{K4}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h04} + {h13} + {h14} + {h33} + {h34} + {v11} + {v31} + {z21} +} \\{{z31} + {z33} + {z42} + {z34} + {z43} + {z36} + {z45} + {z37} + {z47} + {z48}} \\{{h01} + {h03} + {h14} + {h34} + {v12} + {v32} + {z22} + {z32} + {z41} +} \\{{z34} + {z43} + {z35} + {z44} + {z37} + {z46} + {z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h04} + {h31} + {v13} + {v33} + {z31} + {z23} +} \\{{z33} + {z42} + {z44} + {z36} + {z45} + {z38} + {z47}}\end{matrix} \\{{h01} + {h03} + {h12} + {h04} + {h13} + {h14} + {h32} + {h33} + {h34} +} \\{{v14} + {v34} + {z32} + {z41} + {z24} + {z33} + {z42} + {z35} + {z36} +} \\{{z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h21} + {h04} + {h22} + {h31} + {h23} +} \\{{h34} + {v15} + {v35} + {z32} + {z41} + {z33} + {z42} + {z25} + {z35} +} \\{{z44} + {z36} + {z46} + {z38} + {z47}}\end{matrix} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h22} + {h31} + {h23} + {h32} +} \\{{h24} + {v16} + {z31} + {v36} + {z41} + {z33} + {z42} + {z34} + {z43} +} \\{{z26} + {z35} + {z36} + {z45} + {z37} + {z47} + {z48}}\end{matrix} \\{{h03} + {h04} + {h13} + {h22} + {h23} + {h32} + {h24} + {h33} + {v17} +} \\{{z32} + {z41} + {v37} + {z42} + {z34} + {z43} + {z35} + {z44} + {z27} +} \\{{z36} + {z37} + {z46} + {z38} + {z48}} \\{{h01} + {h02} + {h03} + {h21} + {h22} + {h14} + {h24} + {h33} +} \\{{z31} + {v18} + {z32} + {z41} + {\tau 38} + {z43} + {z35} + {z45} +} \\{{z28} + {z37} + {z46}}\end{pmatrix}} \\{{K5}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v11} + {v31} + {v51} + {z21}} \\{{v12} + {v32} + {v52} + {z22}}\end{matrix} \\{{v13} + {v33} + {v53} + {z23}}\end{matrix} \\{{v14} + {v34} + {v54} + {z24}}\end{matrix} \\{{v15} + {v35} + {v55} + {z25}}\end{matrix} \\{{v16} + {v36} + {\upsilon 56} + {z26}}\end{matrix} \\{{v17} + {v37} + {v57} + {z27}}\end{matrix} \\{{v18} + {v38} + {v58} + {z28}}\end{pmatrix}} \\{{K5}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h21} + {h13} + {h22} + {h23} + {v51} + {z33} + {z34} +} \\{{z43} + {z44} + {z37}} \\{{h11} + {h03} + {h21} + {h22} + {h14} + {h23} + {h24} + {v52} +} \\{{z34} + {z35} + {z44} + {z38}}\end{matrix} \\{{h01} + {h11} + {h12} + {h04} + {h22} + {h23} + {h24} + {v53} +} \\{{z31} + {z41} + {z35} + {z36}}\end{matrix} \\{{h01} + {h12} + {h21} + {h22} + {h24} + {v54} + {z32} + {z33} +} \\{{z42} + {z34} + {z43} + {z44} + {z36}}\end{matrix} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h13} + {h22} + {h14} +} \\{{h33} + {h34} + {v55} + {z33} + {z36} + {z46}}\end{matrix} \\{{h03} + {h04} + {h13} + {h22} + {h14} + {h23} + {h34} + {z31} +} \\{{v56} + {z34} + {z37} + {z47}}\end{matrix} \\{{h21} + {h04} + {h31} + {h14} + {h23} + {h24} + {z31} + {z32} +} \\{{\upsilon 57} + {z35} + {z45} + {z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h21} + {h04} + {h13} +} \\{{h14} + {h32} + {h24} + {h33} + {h34} + {z32} + {z35} + {v58} + {z45}}\end{pmatrix}} \\{{K5}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h12} + {h04} + {h13} + {h23} + {h32} + {h24} + {h34} +} \\{{v51} + {z43} + {z44} + {z47}} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h24} + {h33} +} \\{{v52} + {z44} + {z45} + {z48}}\end{matrix} \\{{h01} + {h02} + {h12} + {h21} + {h04} + {h31} + {h14} + {h32} +} \\{{h34} + {v53} + {z41} + {z45} + {z46}}\end{matrix} \\{{h01} + {h11} + {h03} + {h12} + {h04} + {h22} + {h31} + {h23} +} \\{{h24} + {h33} + {h34} + {v54} + {z42} + {z43} + {z44} + {z46}}\end{matrix} \\{{h11} + {h21} + {h14} + {h34} + {v55} + {z43} + {z46}}\end{matrix} \\{{h11} + {h12} + {h22} + {h31} + {z41} + {v56} + {z44} + {z47}}\end{matrix} \\{{h12} + {h13} + {h23} + {h32} + {z41} + {z42} + {v57} + {z45} + {z48}}\end{matrix} \\{{h13} + {h24} + {h33} + {h34} + {z42} + {v58} + {z45}}\end{pmatrix}} \\{{K5}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h12} + {h04} + {h13} + {h23} + {h32} + {h24} + {h34} +} \\{{v\quad 01} + {v21} + {v41} + {z11} + {z43} + {z44} + {z47}} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h24} + {h33} +} \\{{v02} + {v22} + {v42} + {z12} + {z44} + {z45} + {z48}}\end{matrix} \\{{h01} + {h02} + {h12} + {h21} + {h04} + {h31} + {h14} + {h32} +} \\{{h34} + {v03} + {v23} + {v43} + {z13} + {z41} + {z45} + {z46}}\end{matrix} \\{{h01} + {h11} + {h03} + {h12} + {h04} + {h22} + {h31} + {h23} + {h24} +} \\{{h33} + {h34} + {v04} + {v24} + {v44} + {z14} + {z42} + {z43} + {z44} + {z46}}\end{matrix} \\{{h11} + {h21} + {h14} + {h34} + {v05} + {v25} + {v45} + {z15} + {v43} + {z46}} \\{{h11} + {h12} + {h22} + {h31} + {v06} + {v26} + {z41} + {v46} + {z16} +}\end{matrix} \\{{z44} + {z47}}\end{matrix} \\{{h12} + {h13} + {h23} + {h32} + {v07} + {v27} + {z41} + {z42} + {v47} +}\end{matrix} \\{{z17} + {z45} + {z48}} \\{{h13} + {h24} + {h33} + {h34} + {v08} + {v28} + {z42} + {v48} + {z18} + {z45}}\end{pmatrix}} \\{{K6}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h21} + {h13} + {h22} + {h23} + {v11} + {v31} + {v51} + {v61} + {z21} +} \\{{z33} + {z34} + {z43} + {z44} + {z37}} \\{{h11} + {h03} + {h21} + {h22} + {h14} + {h23} + {h24} + {v12} + {v32} + {v52} +} \\{{v62} + {z22} + {z34} + {z35} + {z44} + {z38}}\end{matrix} \\{{h01} + {h11} + {h12} + {h04} + {h22} + {h23} + {h24} + {v13} + {v33} + {v53} +} \\{{z31} + {v63} + {z23} + {z41} + {z35} + {z36}}\end{matrix} \\{{h01} + {h12} + {h21} + {h22} + {h24} + {v14} + {v34} + {v54} + {z32} + {v64} +} \\{{z24} + {z33} + {z42} + {z34} + {z43} + {z44} + {z36}}\end{matrix} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h13} + {h22} + {h14} + {h33} + {h34} +} \\{{v15} + {v35} + {v55} + {z33} + {v65} + {z25} + {z36} + {z46}}\end{matrix} \\{{h03} + {h04} + {h13} + {h22} + {h14} + {h23} + {h34} + {v16} + {z31} + {v36} +} \\{{v56} + {z34} + {v66} + {z26} + {z37} + {z47}}\end{matrix} \\{{h21} + {h04} + {h31} + {h14} + {h23} + {h24} + {v17} + {z31} + {z32} + {v37} +} \\{{v57} + {z35} + {v67} + {z27} + {z45} + {z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h04} + {h13} + {h14} + {h32} + {h24} +} \\{{h33} + {h34} + {v18} + {z32} + {v38} + {z35} + {v58} + {z45} + {v68} + {z28}}\end{pmatrix}} \\{{K6}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h03} + {h12} + {h04} + {h14} + {h23} + {v61} + {z32} +} \\{{z33} + {z42} + {z43} + {z35} + {z36} + {z38}} \\{{h02} + {h11} + {h03} + {h12} + {h21} + {h04} + {h13} + {h24} + {v62} + {z31} +} \\{{z41} + {z33} + {z34} + {z43} + {z35} + {z44} + {z36} + {z37}}\end{matrix} \\{{h11} + {h03} + {h12} + {h04} + {h13} + {h22} + {h14} + {v63} + {z32} + {z42} +} \\{{z34} + {z35} + {z44} + {z36} + {z37} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h13} + {h22} + {z31} + {z32} + {z41} + {v64} +} \\{{z42} + {z35} + {z37}}\end{matrix} \\{{h02} + {h12} + {h22} + {h23} + {h33} + {z31} + {z33} + {v65} + {z34} + {z36} +} \\{{z37} + {z46} + {z47}}\end{matrix} \\{{h03} + {h21} + {h13} + {h31} + {h23} + {h24} + {h34} + {z32} + {z34} + {v66} +} \\{{z35} + {z45} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h11} + {h04} + {h22} + {h31} + {h14} + {h32} + {h24} + {z31} + {z33} +} \\{{v67} + {z36} + {z46} + {z38} + {z48}}\end{matrix} \\{{h01} + {h11} + {h21} + {h22} + {h32} + {z32} + {z33} + {z35} + {z36} + {z45} +} \\{{v68} + {z46}}\end{pmatrix}} \\{{K6}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h12} + {h04} + {h13} + {h22} + {h14} + {h23} + {h34} + {v61} + {z42} + {z43} +} \\{{z45} + {z46} + {z48}} \\{{h01} + {h21} + {h13} + {h31} + {h14} + {h23} + {h24} + {v62} + {z41} + {z43} +} \\{{z44} + {z45} + {z46} + {z47}}\end{matrix} \\{{h02} + {h22} + {h14} + {h32} + {h24} + {v63} + {z42} + {z44} + {z45} + {z46} +} \\{{z47} + {z48}}\end{matrix} \\{{h11} + {h03} + {h12} + {h21} + {h04} + {h13} + {h22} + {h14} + {h33} + {h34} +} \\{{z41} + {v64} + {z42} + {z45} + {z47}}\end{matrix} \\{{h12} + {h21} + {h22} + {h31} + {h14} + {h34} + {z41} + {v65} + {z43} + {z44} +} \\{{z46} + {z47}}\end{matrix} \\{{h11} + {h13} + {h22} + {h31} + {h23} + {h32} + {z42} + {v66} + {z44} + {z45} +} \\{{z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h21} + {h14} + {h23} + {h32} + {h24} + {h33} + {z41} + {z43} +} \\{{v67} + {z46} + {z48}}\end{matrix} \\{{h11} + {h21} + {h13} + {h14} + {h24} + {h33} + {z42} + {z43} + {z45} + {v68} +} \\{z46}\end{pmatrix}} \\{{K6}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h12} + {h04} + {h13} + {h22} + {h14} + {h23} + {h34} + {v01} + {v21} + {v41} +} \\{{z11} + {z42} + {z43} + {z45} + {z46} + {z48}} \\{{h01} + {h21} + {h13} + {h31} + {h14} + {h23} + {h24} + {v02} + {v22} + {v42} +} \\{{z12} + {z14} + {z43} + {z44} + {z45} + {z46} + {z47}}\end{matrix} \\{{h02} + {h22} + {h14} + {h32} + {h24} + {v03} + {v23} + {z13} + {z42} + {z44} +} \\{{z45} + {z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h03} + {h12} + {h21} + {h04} + {h13} + {h22} + {h14} + {h33} + {h34} +} \\{{v04} + {v24} + {v44} + {z14} + {z41} + {z42} + {z45} + {z47}}\end{matrix} \\{{h12} + {h21} + {h22} + {h31} + {h14} + {h34} + {v05} + {v25} + {v45} + {z41} +} \\{{z15} + {z43} + {z44} + {z46} + {z47}}\end{matrix} \\{{h11} + {h13} + {h22} + {h31} + {h23} + {h32} + {v06} + {v25} + {v46} + {z42} +} \\{{z16} + {z44} + {z45} + {z47} + {z48}}\end{matrix} \\{{h11} + {h12} + {h21} + {h14} + {h23} + {h32} + {h24} + {h33} + {v07} + {v27} +} \\{{z41} + {v47} + {z43} + {z17} + {z46} + {z48}}\end{matrix} \\{{h11} + {h21} + {h13} + {h14} + {h24} + {h33} + {v08} + {v28} + {z42} + {z43} +} \\{{v48} + {z18} + {z45} + {z46}}\end{pmatrix}} \\{{K7}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {h12} + {h13} + {h23} + {h33} + {h34} + {v01} + {v21} +} \\{{v41} + {z11} + {v71} + {z31} + {z32} + {z34} + {z35} + {z45} + {z37} + {z38} +} \\{{z47} + {z48}} \\{{h01} + {h02} + {h11} + {h12} + {h21} + {h04} + {h13} + {h14} + {h24} + {h34} +} \\{{v02} + {v22} + {v42} + {z12} + {z31} + {v72} + {z32} + {z33} + {z36} + {z46} +} \\{{z38} + {z48}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h13} + {h22} + {h31} + {h14} + {v03} +} \\{{v23} + {v43} + {z13} + {z31} + {z32} + {v73} + {z33} + {z34} + {z35} + {z45} +} \\{{z37} + {z47}}\end{matrix} \\{{h02} + {h11} + {h12} + {h04} + {h22} + {h14} + {h32} + {h33} + {h34} + {v04} +} \\{{v24} + {v44} + {z31} + {z14} + {z33} + {v74} + {z36} + {z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h11} + {h03} + {h12} + {h21} + {h04} + {h31} + {h33} + {h34} + {v05} +} \\{{v25} + {z31} + {v45} + {z32} + {z41} + {z15} + {z42} + {z34} + {v75} + {z35} +} \\{{z44} + {z37} + {z38}}\end{matrix} \\{{h02} + {h12} + {h04} + {h13} + {h22} + {h32} + {h34} + {v06} + {v26} + {z31} +} \\{{z32} + {z41} + {v46} + {z33} + {z42} + {z16} + {z43} + {v76} + {z36} + {z38}}\end{matrix} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h23} + {h33} + {v07} + {z31} +} \\{{v27} + {z32} + {z41} + {z33} + {z42} + {v47} + {z34} + {z43} + {z17} + {z35} +} \\{{z44} + {v77} + {z37}}\end{matrix} \\{{h02} + {h11} + {h03} + {h14} + {h32} + {h24} + {h33} + {v08} + {z31} + {z41} +} \\{{v28} + {z33} + {z43} + {v48} + {z18} + {z36} + {z37} + {v78}}\end{pmatrix}} \\{{K7}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h21} + {h13} + {h32} + {h33} + {v71} + {z32} + {z37} + {z38} + {z47} +} \\{z48} \\{{h02} + {h11} + {h22} + {h31} + {h14} + {h33} + {h34} + {v72} + {z33} + {z38} +} \\{z48} \\{{h11} + {h03} + {h12} + {h23} + {h32} + {h34} + {z31} + {v73} + {z34} + {z35} +} \\{z45}\end{matrix} \\{{h12} + {h04} + {h31} + {h32} + {h24} + {z31} + {v74} + {z36} + {z37} + {z46} +} \\{{z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h13} + {h22} + {h31} + {h32} + {h33} +} \\{{z33} + {z34} + {z43} + {v75} + {z35} + {z44} + {z36} + {z37}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h04} + {h13} + {h22} + {h31} + {h14} + {h23} +} \\{{h32} + {h33} + {h34} + {z34} + {z35} + {z44} + {v76} + {z36} + {z37} + {z38}}\end{matrix} \\{{h02} + {h03} + {h12} + {h21} + {h04} + {h14} + {h23} + {h32} + {h24} + {h33} +} \\{{h34} + {z31} + {z41} + {z36} + {v77} + {z37} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h12} + {h21} + {h04} + {h31} + {h32} + {h24} + {h34} +}\end{matrix} \\{{z32} + {z33} + {z42} + {z34} + {z43} + {z35} + {z44} + {z36} + {v78} + {z38}}\end{pmatrix}} \\{{K7}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h12} + {h04} + {h13} + {h32} + {h33} + {v71} + {z33} + {z42} + {z34} + {z43} +} \\{{z44} + {z36} + {z37} + {z38} + {z47} + {z48}} \\{{h01} + {h11} + {h13} + {h31} + {h14} + {h33} + {h34} + {v72} + {z34} + {z43} +} \\{{z44} + {z37} + {z38} + {z48}}\end{matrix} \\{{h02} + {h12} + {h14} + {h32} + {h34} + {z31} + {v73} + {z44} + {z45} + {z38}} \\{{h11} + {h03} + {h12} + {h04} + {h31} + {h32} + {z32} + {z41} + {z33} + {z42} +} \\{{v74} + {z34} + {z43} + {z35} + {z44} + {z36} + {z37} + {z46} + {z38} + {z47} +} \\{z48}\end{matrix} \\{{h01} + {h11} + {h12} + {h32} + {h24} + {h34} + {z31} + {z32} + {z34} + {z43} +} \\{{v75} + {z44} + {z45} + {z37} + {z46} + {z38} + {z48}}\end{matrix} \\{{h02} + {h12} + {h21} + {h13} + {h31} + {h33} + {z31} + {z32} + {z33} + {z44} +} \\{{v76} + {z45} + {z46} + {z38} + {z47}}\end{matrix} \\{{h11} + {h03} + {h13} + {h22} + {h31} + {h14} + {h32} + {h34} + {z31} + {z32} +} \\{{z41} + {z33} + {z34} + {z35} + {z45} + {v77} + {z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h04} + {h31} + {h14} + {h23} + {h24} + {h33} + {h34} + {z31} + {z33} +}\end{matrix} \\{{z42} + {z43} + {z44} + {z36} + {z45} + {z37} + {v78} + {z38} + {z47}}\end{pmatrix}} \\{{K7}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h12} + {h04} + {h13} + {h32} + {h33} + {v11} + {v31} + {v51} + {v61} + {z21} +} \\{{z33} + {z42} + {z34} + {z43} + {z44} + {z36} + {z37} + {z38} + {z47} + {z48}} \\{{h01} + {h11} + {h13} + {h31} + {h14} + {h33} + {h34} + {v12} + {v32} + {v52} +} \\{{v62} + {z22} + {z34} + {z43} + {z44} + {z37} + {z38} + {z48}}\end{matrix} \\{{h02} + {h12} + {h14} + {h32} + {h34} + {v13} + {v33} + {v53} + {z31} + {v63} +} \\{{z23} + {z44} + {z45} + {z38}}\end{matrix} \\{{h11} + {h03} + {h12} + {h04} + {h31} + {h32} + {v14} + {v34} + {v54} + {z32} +} \\{{z41} + {v64} + {z24} + {z33} + {z42} + {z34} + {z43} + {z35} + {z44} + {z36} +} \\{{z37} + {z46} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h11} + {h12} + {h32} + {h24} + {h34} + {v15} + {v35} + {z31} + {z32} +} \\{{v55} + {v65} + {z25} + {z34} + {z43} + {z44} + {z45} + {z37} + {z46} + {z38} +} \\{z48}\end{matrix} \\{{h02} + {h12} + {h21} + {h13} + {h31} + {h33} + {v16} + {z31} + {v36} + {z32} +} \\{{z33} + {v56} + {v66} + {z26} + {z44} + {z45} + {z46} + {z38} + {z47}}\end{matrix} \\{{h11} + {h03} + {h13} + {h22} + {h31} + {h14} + {h32} + {h34} + {v17} + {z31} +} \\{{z32} + {z41} + {v37} + {z33} + {z34} + {v57} + {z35} + {v67} + {z27} + {z45} +} \\{{z46} + {z47} + {z48}}\end{matrix} \\{{h11} + {h04} + {h31} + {h14} + {h23} + {h24} + {h33} + {h34} + {z31} + {v18} +} \\{{z33} + {z42} + {v38} + {z43} + {z44} + {v58} + {z36} + {z45} + {v68} + {z28} +} \\{{z37} + {z38} + {z47}}\end{pmatrix}} \\{{K8}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h03} + {h04} + {h13} + {h22} + {h31} + {h32} + {h33} + {v11} + {v31} +} \\{{v51} + {v61} + {z21} + {v81} + {z32} + {z42} + {z35} + {z37}} \\{{h11} + {h03} + {h04} + {h31} + {h14} + {h23} + {h32} + {h33} + {h34} + {v12} +} \\{{v32} + {v52} + {v62} + {z22} + {v82} + {z33} + {z43} + {z35} + {z36} + {z38}}\end{matrix} \\{{h11} + {h12} + {h21} + {h04} + {h32} + {h24} + {h33} + {h34} + {v13} + {v33} +} \\{{v53} + {z31} + {v63} + {z23} + {z41} + {v83} + {z34} + {z35} + {z44} + {z36} + {z37}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h04} + {h31} + {h32} + {h34} + {v14} +} \\{{v34} + {z31} + {v54} + {z41} + {v64} + {z24} + {v84} + {z36} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h22} + {h31} + {h32} + {h24} + {h34} + {v15} +} \\{{v35} + {z31} + {v55} + {z33} + {v65} + {z25} + {z35} + {v85} + {z36} + {z45} +} \\{{z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h03} + {h12} + {h21} + {h04} + {h31} + {h23} + {h32} + {h33} +} \\{{v16} + {z31} + {v36} + {z32} + {v56} + {z34} + {v66} + {z26} + {z35} + {z36} +} \\{{z45} + {v86} + {z37} + {z46} + {z38} + {z47} + {z48}}\end{matrix} \\{{h02} + {h03} + {h21} + {h04} + {h13} + {h22} + {h31} + {h32} + {h24} + {h33} +} \\{{h34} + {v17} + {z31} + {z32} + {v37} + {z33} + {v57} + {v67} + {z27} + {z36} +} \\{{z37} + {z46} + {v87} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h02} + {h21} + {h04} + {h31} + {h14} + {h23} + {h24} + {h33} + {v18} +} \\{{z32} + {v38} + {z34} + {z35} + {v58} + {z36} + {z45} + {v68} + {z28} + {z46} +} \\{{z38} + {v88} + {z48}}\end{pmatrix}} \\{{K8}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h02} + {h11} + {h21} + {h23} + {h33} + {z31} + {v81} + {z41} + {z33} +} \\{{z43} + {z35}} \\{{h02} + {h03} + {h12} + {h21} + {h22} + {h31} + {h24} + {h34} + {z31} + {z32} +} \\{{z41} + {v82} + {z42} + {z34} + {z44} + {z36}}\end{matrix} \\{{h01} + {h03} + {h21} + {h04} + {h13} + {h22} + {h31} + {h23} + {h32} + {z31} +} \\{{z32} + {z41} + {z33} + {z42} + {v83} + {z43} + {z37}}\end{matrix} \\{{h01} + {h04} + {h22} + {h14} + {h32} + {h24} + {z32} + {z42} + {z34} + {v84} +} \\{{z44} + {z38}}\end{matrix} \\{{h11} + {h12} + {h21} + {h04} + {h22} + {h31} + {h23} + {h32} + {h24} + {h33} +} \\{{h34} + {v85} + {z38} + {z48}}\end{matrix} \\{{h01} + {h12} + {h13} + {h22} + {h23} + {h32} + {h24} + {h33} + {h34} + {z33} +} \\{{z34} + {z35} + {z45} + {v86}}\end{matrix} \\{{h02} + {h11} + {h13} + {h14} + {h23} + {h24} + {h33} + {h34} + {z34} + {z36} +} \\{{z46} + {v87}}\end{matrix} \\{{h11} + {h03} + {h21} + {h04} + {h22} + {h31} + {h14} + {h23} + {h32} + {h33} +} \\{{z31} + {z32} + {z33} + {z34} + {z37} + {z38} + {z47} + {v88} + {z48}}\end{pmatrix}} \\{{K8}_{3} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h21} + {h04} + {h22} + {h23} + {h32} + {h24} + {h34} + {v81} + {z41} +} \\{{z43} + {z35} + {z36} + {z37} + {z46} + {z47}} \\{{h01} + {h02} + {h22} + {h31} + {h23} + {h24} + {h33} + {z41} + {v82} + {z42} +} \\{{z35} + {z44} + {z36} + {z45} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h02} + {h03} + {h31} + {h23} + {h32} + {h24} + {h34} + {z41} + {z42} + {v83} +} \\{{z43} + {z36} + {z37} + {z46} + {z38} + {z48}}\end{matrix} \\{{h03} + {h21} + {h22} + {h31} + {h23} + {h33} + {h34} + {z42} + {v84} + {z35} +} \\{{z44} + {z36} + {z45} + {z46} + {z38}}\end{matrix} \\{{h22} + {h14} + {h24} + {h33} + {z32} + {z34} + {z43} + {v85} + {z48}}\end{matrix} \\{{h11} + {h21} + {h31} + {h23} + {h34} + {z31} + {z41} + {z33} + {z44} + {z45} +} \\{v86}\end{matrix} \\{{h12} + {h21} + {h22} + {h31} + {h32} + {h24} + {z31} + {z32} + {z41} + {z42} +} \\{{z34} + {z46} + {v87}}\end{matrix} \\{{h21} + {h13} + {h14} + {h23} + {h32} + {h24} + {z31} + {z33} + {z42} + {z34} +} \\{{z47} + {v88} + {z48}}\end{pmatrix}} \\{{K8}_{4} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h21} + {h04} + {h22} + {h23} + {h32} + {h24} + {h34} + {v01} + {v21} +} \\{{v41} + {z11} + {v71} + {z41} + {z43} + {z35} + {z30} + {z37} + {z46} + {z47}} \\{{h01} + {h02} + {h22} + {h31} + {h23} + {h24} + {h33} + {v02} + {v22} + {v42} +} \\{{z12} + {v72} + {z41} + {z42} + {z35} + {z44} + {z36} + {z45} + {z37} + {z38} +} \\{{z47} + {z48}}\end{matrix} \\{{h02} + {h03} + {h31} + {h23} + {h32} + {h24} + {h34} + {v03} + {v23} + {v43} +} \\{{z13} + {z41} + {v73} + {z42} + {z43} + {z38} + {z37} + {z46} + {z38} + {z48}}\end{matrix} \\{{h03} + {h21} + {h22} + {h31} + {h23} + {h33} + {h34} + {v04} + {v24} + {v44} +} \\{{z14} + {z42} + {v74} + {z35} + {z44} + {z36} + {z45} + {z46} + {z38}}\end{matrix} \\{{h22} + {h14} + {h24} + {h33} + {v05} + {v25} + {v45} + {z32} + {z15} + {z34} +} \\{{z43} + {v75} + {z48}}\end{matrix} \\{{h11} + {h21} + {h31} + {h23} + {h34} + {v06} + {v26} + {z31} + {z41} + {v46} +} \\{{z33} + {z16} + {z44} + {v76} + {z45}}\end{matrix} \\{{h12} + {h21} + {h22} + {h31} + {h32} + {h24} + {v07} + {z31} + {v27} + {z32} +} \\{{z41} + {z42} + {v47} + {z34} + {z17} + {v77} + {z46}}\end{matrix} \\{{h21} + {h13} + {h14} + {h23} + {h32} + {h24} + {v08} + {z31} + {v28} + {z33} +} \\{{z42} + {z34} + {v48} + {z18} + {v78} + {z47} + {z48}}\end{pmatrix}} \\{{K9}_{1} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h02} + {h12} + {h21} + {h31} + {h23} + {v01} + {v21} + {v41} + {z11} + {v71} +} \\{{z31} + {z32} + {v91} + {z34} + {z36} + {z37} + {z46} + {z38} + {z47} + {z48}} \\{{h03} + {h21} + {h13} + {h22} + {h32} + {h24} + {v02} + {v22} + {v42} + {z12} +} \\{{z31} + {v72} + {z32} + {z33} + {v92} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h01} + {h11} + {h21} + {h04} + {h22} + {h14} + {h23} + {h33} + {v23} + {v43} +} \\{{z13} + {z31} + {z32} + {v73} + {z33} + {z34} + {v93} + {z38} + {z48}}\end{matrix} \\{{h01} + {h11} + {h22} + {h24} + {h34} + {v04} + {v24} + {v44} + {z31} + {z14} +} \\{{z33} + {v74} + {z35} + {v94} + {z36} + {z45} + {z37} + {z46} + {z38} + {z47} +} \\{z48}\end{matrix} \\{{h01} + {h02} + {h12} + {h04} + {h31} + {h14} + {h32} + {h33} + {h34} + {v05} +} \\{{v25} + {z31} + {v45} + {z41} + {z15} + {v75} + {z35} + {v95} + {z38}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h13} + {h32} + {h33} + {h34} + {v06} + {v26} +} \\{{z32} + {v46} + {z42} + {z16} + {z35} + {v76} + {z36} + {v96}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h04} + {h14} + {h33} + {h34} + {v07} +} \\{{v27} + {z33} + {v47} + {z43} + {z17} + {z36} + {v77} + {z37} + {v97}}\end{matrix} \\{{h01} + {h11} + {h03} + {h13} + {h31} + {h14} + {h32} + {h33} + {v08} + {v28} +} \\{{z34} + {v48} + {z44} + {z18} + {z37} + {v78} + {v98}}\end{pmatrix}} \\{{K9}_{2} = \begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h01} + {h11} + {h03} + {h21} + {h13} + {z32} + {v91} + {z35} + {z36} + {z45} +} \\{z46} \\{{h01} + {h02} + {h11} + {h12} + {h04} + {h22} + {h14} + {z33} + {v92} + {z36} +} \\{{z37} + {z46} + {z47}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h13} + {h23} + {z31} + {z34} + {v93} +} \\{{z35} + {z45} + {z37} + {z38} + {z47} + {z48}}\end{matrix} \\{{h02} + {h12} + {h04} + {h14} + {h24} + {z31} + {z35} + {v94} + {z45} + {z38} +} \\{z48}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h04} + {h13} + {h31} + {h14} + {z31} +} \\{{z32} + {z41} + {z42} + {z36} + {v95} + {z38}}\end{matrix} \\{{h02} + {h03} + {h12} + {h04} + {h13} + {h14} + {h32} + {z32} + {z33} + {z42} +} \\{{z43} + {z35} + {z37} + {v96}}\end{matrix} \\{{h03} + {h04} + {h13} + {h14} + {h33} + {z31} + {z41} + {z33} + {z34} + {z43} +} \\{{z35} + {z44} + {z36} + {z38} + {v97}}\end{matrix} \\{{h01} + {h02} + {h11} + {h03} + {h12} + {h13} + {h34} + {z31} + {z41} + {z34} +} \\{{z35} + {z44} + {z37} + {z38} + {v98}}\end{pmatrix}}\end{matrix}$

[0132] Then, the next step S103 is executed to carry out a variabletransposition process. With the results of the vectors K11, K12, K13,K14, K21, . . . , K91 and K92 used as a base, the simultaneous linearequation is transformed so as to result in equations, which each includeonly terms zxx and vxx on the right-hand side thereof as follows.

k1₁₁ =v11+z21

k1₁₂ =v12+z22

k1₁₃ =v13+z23

k1₁₄ =v14+z24

k1₁₅ =v15+z25

k1₁₆ =v16+z26

k1₁₇ =v17+z27

k1₁₈ =v18+z28

h11+h21+h13 30 k1₂₁ =v11+z32+z42

h11+h12+h22+h14+k1₂₂ =v12+z33+z43

h11+h12+h13+h23+k1₂₃ =v13+z31+z41+z34+z44

h12+h14+h24+k1₂₄ =v14+v31+z41

h01+h02+h03+h04+h31+k1₂₅ =v15+z36+z46+z38+z48

h02+h03+h04+h32+k1₂₆ =v16+z35+z45+z37+z47

h03+h04+h33k1₂₇ =v17+z35+z36+z45+z46+z38+z48

h01+h02h03+h34+k1₂₈ =v18+z35+z45+z37+z38+z47+z48

h01+h03+k1₃₁ =v11+z42+z35+z36z45+z46

h01+h02+h04+k1₃₂ =v12+z43+z36+z37+z46+z47

h01+h02+h03+k1₃₃ =v13+z41+z35+z44+z45+z37+z38+z47+z48

h02+h04+k1₃₄ =v14+z41+z35+z45+z38+z48

h11+h12+h13+h14+k1₃₅ =v15+z31+z32+z41+z42+z46+z48

h12+h13+h14+k1₃₆ =v16+z32+z33+z42+z43+z45+z47

h13+h14+k1₃₇ =v17+z31+z41+z33+z34+z43+z44+z45+z46+z48

h11+h12+h13+z31+k1₃₈ =v18+z41+z3431+z44+z45+z47+z48

h01+h03+k1₄₁ =v01+z11+z42+z35+z36+z45+z46

h01+h02+h04+k1₄₂ =v02+z12+z43+z36+z37+z46+z47

h01+h02+h03+k1₄₃ =v03+z13+z41+z35+z44+z45+z37+z38+z47+z48

h02+h04+k1₄₄ =v04+z14+z41+z35+z45+z38+z48

h11+h12+h13+h14+k1₄₅ =v05+z31+z41+z15+z42+z46+z48

h12+h13h14+k1₃₆ =v06+z32+z33+z42+z16+z43+z45+z47

h13+h14+k1₄₇ =v07+z31+z41+z33+z34+z43+z17+z44+z45+z46+z48

h11+h12+h13+k1₄₈ =v08+z31+z41+z34+z44+z18+z45+z47+z48

k2₁₁ =v01+v21+z11

k2₁₂ =v02+v22+z12

k2₁₃ =v03+v23+z13

k2₁₄ =v04+v31+z14

k2₁₅ =v05+v25+z15

k2₁₆ =v06+v26+z16

k2₁₇ =v07+v27+z17

k2₁₈ =v08+v28+z18

h02+h12+h21+h31h23+k2₂₁ =v21+z31+z32+z34+z36+z37+z46+z38+z47+z48

h03+h21+h13+h22h24+k2₂₂ =v22+z31+z32+z33+z37+z38+z47+z48

h01+h11+h21 30 h04+h22+h14+h23+h33+k2₂₃ =v23+z31+z32+z33+z34+z38+z48

h01+h11+h22+h24+h34+k2₂₄ =v24+z31+z33+z35+z36+z45+z37+z46+z38+z47+z48

h01+h02+h12+h04+h31+h32+h33+h34+k2₂₅ =v25+z31+z41+z35+z38

h01+h02+h11+h03+h13+h32 30 h33+h34+k2₂₆ =v26+z32+z42+z35+z36

h01+h02+h11+h03+h12+h04+h14+h33+h34+k2₃₇ =v27+z33+z43+z36+h37

h01+h11+h03+h13+h31+h14+h32+k2₂₈ =v28+z34+z44+z37

h01+h11+h12h1₃₁ 30 h33+k2₃₁ =v21+z32+z41+z33+z34 30z43+z35+z36+z37+z46+z38+z47+z48

h02+h12h13+h31+h32+h34+k2₃₂ =v22+z41+z33+z42+z34+z44+z36+z37+z38+z47+z48

h11+h03h13+h31+h14+h32+h33+k2₃₃ =v23+z41+z42+z34+z43+z37+z38+z48

h11+h04h14+h32+h34+k2₃₄=v24+z31+z32+z33+z42+z34+z35+z44+z36+z45+z37+z46+z47+z48

h01+h02+h11+h21+h22+h14+h23+h24+k2₃₅ =v25+z31+z32+z41+z33+z35+z48

h02+h11+h03+h12+h22+h23+h24+k2₃₅ =v26+z32+z33+z42+z34+z36+z45

h01+h03+h12+h04+h13+h23+h24+k2₃₇ =v27+z33+z34+z43+z37+z46

h01+h21+h04+h13+h22+h23+z31+z32+k2₂₈ =v28+z33+z44+z38+z47+z48

h01+h11+h12+h31+h33k2₄₁=v11+z21+z32+z33+z41+z33+z34+z43+z35+z36+z37+z46+z38+z47+z48

h02+h12+h13+h31+h32+h34+k2₄₃=v12+z22+z41+z33+z42+z34+z44+z36+z37+z38+z47+z48

h11+h03+h13+h31+h14+h32+h33+k2₄₃=v13+z23+z41+z42+z34+z43+z37+z38+z37+z38+z48

h11+h04+h14+h34+k2₄₄=v14+z31+z32+z24+z33+z42+z34+z35+z44+z36+z45+z37+z46+z47+z48

h01+h02+h11+h21+h22+h14+h23+h24+k2₄₅=v15+z31+z32+z41+z33+z25+z34+z35+z48

h02+h11+h03+h12+h22+h23+h24+k2₄₈ =v16+z32+z33+z42+z34+z26+z36+z45

h01+h03+h12 30 h04+h13+h23+h24+k2₄₇ =v17+z33+z34+z43+z27+z46

h01+h21+h04+h13+h22+h23+z31+k2₄₈ =v18+z32+z33+z44+z28+z38+z47+z48

k3₁₁ =v11+v31+z21

k3₁₂ =v12+v32+z22

k3₁₃ =v13+v33+z23

k3₁₄ =v14+v34+z24

k3₁₅ =v15+v35+z25

k3₁₆ =v16+v36+z26

k3₁₇ =v17+v37+z27

k3₁₈ =v18+v38+z28

h02+h03+h04+h13+h22+h31 30 h33+k3₃₁ =v31+z32+z42+z35+z37

h11+h03+h04+h31+h14+h23+h32+h33+h34+k3₂₂ =v32+z33+z43+z35+h36+z38

h11+h12+h21+h04+h32+h24+h33+h34+k3₂₃ =v33+z31+z41+z34+z35+z44+z36+z37

h01+h02+h03h12+h21 30 h04+h31+h32+h34+k3₂₄ =v34+z31+z41+z36+z38

h01+h02h11+h03+h22+h31+h32+h24+h34+k3₂₅=v35+z31+z33+z35+z36+z45+z37+z46+z47

h01+h02h03+h12+h21+h04+h31+h23+h32+k3₃₃ h31+k3₂₆=v36+z32+z34+z35+z36+z45+z37+z46+z38+z47+z48

h02+h03h21+h04+h13+h22+h31 30 h32+h24+h33+h34+z31+z32+k3₂₇=v37+z33+z36+z37+z46+z38+z47+z48

h01+h02+h21+h04+h31+h14+h23+h24+h33+z32+k3₂₈=v38+z34+z35+z36+z45+z46+z38+z48

h01+h02+h03+h21+h04+h22+h31+k3₃₁ =v31+z42+z35+z38+z47+z48

h02+h03+h04+h22+h23+h32+k3₃₂ =v32+z43+z35+z36+z48

h02+h21+h04+h23+h24+h33+k3₃₂ =v33+z41+z44+z36+z45+z37

h01+h02+h03+h21+h24h34+k3₃₄ =v34+z41+z37+z46+z47+z48

h11+h12+h21+h13+h24+h33+h34+k3₃₅ =v35+z31+z34+z43+z44+z45+z46+z47

h11+h12+h21+h13+h22+h14+h34+z31+k3₃₆ =v36+z32+z44+z45+z46+z47+z48

h12+h13+h22+h31+h14+z32+z41+k3₃₇ =v37+z33+z46+z47+z48

h11+h12+h14 30 h23+h32+h33+h34+z33+z42+k3₃₈ =v38+z43+z44+z45+z46+z48

h01+h02+h03+h21+h04+h22+h31+k3₄₁ =v01+v21+z11+z42+z35+z38+z47+z48

h02+h03+h04+h22+h23+h32+k3₄₂ =v02+v22+z12+z43+z35+z36+z48

h03+h21+h04+h23+h24+h33+k3₄₃ =v03+v23+z13+z41+z44+z36+z45+z37

h01+h02+h03+h21+h24+h34+k3₄₄ =v04+v24+z14+z41+z37+z46+z47+z48

h11+h12+h21+h13+h24+h33+h34+k3₄₅=v05+v25+z31+z15+z34+z43+z44+z45+z46+z47

h11+h12+h21+h13+h24+h33+h34+k3₄₅=v05+v25+z31+z15+z34+z43+z44+z45+z46+z48

h12+h13+h22+h31+h14+h23+k3₄₇ =v07+v27+z32+z41+z33+z17+z46+z47+z46+z48

h11+h12+h14+h23+h32+h33+h34+k3₄₈=v08+v28+z33+z42+z43+z44+z18+z45+z46+z48

k4₁₁ =v01+v41+z11

k4₁₂ =v02+v42+z12

k4₁₃ =v03+v43+z13

k4₁₄ =v04+v44+z14

k4₁₅ =v05+v45+z15

k4₁₆ =v06+v46+z16

k4₁₇ =v07+v47+z17

k4₁₈ =v08+v48+z18

h01+h11+h03+h12+h13+h23 30 h33+h34+k4₄₁=v41+z31+z32+z34+z35+z45+z37+z38+z47+z48

h01+h02+h11+h12+h21+h04+h13+h14+h24+h34+k4₂₂=v42+z31+z32+z33+h36+z46+z38+z48

h01+h02+h03+h12+h21+h13+h22+h31+h14+k4₂₃=v43+z31+z32+z33+z34+z35+z45+z37+z47

h02+h11+h12+h04+h22 30 h14+h32+h33+h34+k4₂₄ =v44+z31+z33+z36+z37+z46+z47

h01+h02h11+h03+h22+h31+h32+h24+h34+k3₂₅=v35+z31+z33+z35+z36+z45+z37+z46+z47

h01+h11+h03+h12+h21+h04+h31+h33+h34+z31+k4₂₅v45+z32+z41+z42+z34+z35+z44+z37+z38

h02+h12+h04+h13+h22+h32+h34 30 z31+z32+z41+k4₂₆ =v46+z33+z42+z43+z36+z38

h01+h11+h03 30 h13+h31+h14+h23+h33+z31+z32+z41+z33+z42+k4₂₇=v47+z34+z43+z35+z44+z37

h02+h11+h03+h14+h32+h24+h33+z31+z41+z33+z43+k4₂₈ =v48+v36+z37

h02+h04+h13+h14+h33+h34+k4₃₁=v41+z31+z33+z42+z34+z43+z36+z45+z37+z47+z48

h01+h03+h14+h34+k4₃₂ =v42+z32+z41+z34+z43+z35+z44+z37+z46+z38+z48

h01+h02+h11+h04+h31+k4₃₈ =v43+z31+z33+z42+z44+z36+z45+z38+z47

h01+h03+h12+h04+h13+h14+h32+h33+h34+k4₃₄=v44+z32+z41+z33+z42+z35+z36+z46+z47

h01+h02+h11+h03+h21+h04+h22+h31+h23+h34+k4₃₅=v45+z32+z41+z33+z42+z35+z44+z36+z46+z38+z47

h02+h03+h12+h21+h04+h22+h31+h23+h24+z31+z41+k4₃₆=v46+z33+z42+z34+z43+z35+z36+z45+z37+z47+z48

h03+h04+h13+h22+h23+h32+h24+h33+z32+z41+z42+k4₃₇=v47+z34+z43+z35+z44+z36+z37+z46+z38+z48

h01+h02+h03+h21+h22+h14+h24+h33+z31+z32+z41+k4₃₈ =v47+z35+z45+z37+z46

h02+h04+h13+h14+h33+h34+k4₄₃=v11+v31+z21+z31+z33+z42+z34+z43+z36+z45+z37+z47+z48

h01+h03+h14+h34+k4₄₂=v12+v32+z22+z32+z41+z34+z43+z35+z44+z37+z46+z38+z48

h01+h02+h11+h04+h31+k4₄₃ =v13+v33+z31+z23+z33+z42+z44+z36+z45+z38+z47

h01+h03+h12+h04+h13+h14+h32+h33+h34+k4₄₄=v14+v34+z32+z41+z24+z33+z42+z35+z36+z46+z47

h01+h02+h11+h03+h21+h04+h22+h31+h23+h34+k4₄₅=v15+v35+z32+z41+z33+z42+z25+z35+z44+z36+z46+z38+z47

h02+h03+h12+h21+h04+h22+h31+h23+h32+h24+k4₄₆=v16+z31+v36+z41+z33+z42+z34+z43+z26+z35+z36+z45+z37+z47+z48

h03+h04+h13+h22+h23+h32+h24+h33+k4₄₇=v17+z32+z41+v37+z42+z34+z43+z35+z44+z27+z36+z37+z46+z38+z48

h01+h02+h03+h31+h22+h14+h24+h33+z31+k4₄₈=v18+z32+z41+v38+z43+z35+z45+z28+z37+z46

k5₁₁ =v31+v51+z21

k5₁₂ =v32+v52+z22

k5₁₃ =v33+v53+z23

k5₁₄ =v34+v54+z24

k5₁₅ =v35+v55+z25

k5₁₆ =v36+v56+z26

k5₁₇ =v37+v57+z27

k5₁₈ =v38+v58+z28

h02+h21+h13 30 h22+h23+k5₂₁ =v51+z33+z34+z43+z44+z37

h11+h03+h21+h22+h14+h23+h24+k5₂₂ =v52+z34+z35+z44+z38

h01+h11+h12+h04+h22+h24+k5₂₃ =v53+z31+z41+z35+z36

h01+h12+h21+h22+h24+k5₂₄ =v54+z32+z33+z42+z34+z43+z44+z36

h02+h03+h12+h21+h04+h13+h22+h14+h33+h34+k5₂₆ =v55+z33+z36+z46

h03+h04+h13+h22+h14+h23+h34+z31+k5₂₆ =v56+z34+z37+z47

h21+h04+h31+h14+h23+h24+z31+z32+k5₂₇ =v57+z35+z45+z38+z48

h01+h02+h11+h03+h12+h21+h04+h13+h14+h32+h24+h33+h34+z32+z35+k5₂₈=v58+z45

h02+h12+h04+h13+h23+h32+h24+h34+k5₃₁ =v51+z43+z44+z47

h01+h11+h03+h13+h31+h14+h24+h33+k5₃₂ =v52+z44+z45+z48

h01+h02+h12+h21+h04+h31+h14+h32+h34+k5₃₃ =v53+z41+z45+z46

h01+h11+h03+h12+h04+h22+h31+h23+h24+h33+h34+k5₃₄ =v54+z42+z43+z44+z46

h11+h21+h14+h34+k5₃₆ =v55+z43+z46

h11+h12+h22+h31+z41+k5₃₆ =v56+z44+z47

h12+h13+h23+h32+z41+z42+k5₃₇ =v57+z45+z48

h13+h24+h33+h34+z42+k5₃₈ =v58+z45

h02+h12+h04+h13+h23+h32+h24+h34+k5₄₁ =v01+v21+v41+z11+z43+z44+z47

h01+h11+h03+h13+h31+h14+h24+h33+k5₄₂ =v02+v22+v42+z12+z44+z45+z48

h01+h02+h12+h21+h04+h31+h14+h32+h34+k5₄₃ =v03+v23+v43+z13+z41+z45+z46

h01+h11+h03+h12+h04+h22+h31+h23+h24+h33+h34+k5₄₄=v04+v24+v44+z14+z42+z43+z44+z46

h11+h21+h14+h34+k5₄₅ =v05+v25+v45+z15+z43+z46

h11+h12+h22+h31k5₄₆ =v06+v26+z41+z46+z16+z44+z47

h12+h13+h23+h32k5₄₇ =v07+v27+z41+z42+v47+z17+z45+z48

h13+h24+h33+h34k5₄₈ =v08+v28+z42+z48+z18+z45

h02+h21+h13 30 h22+h23+k6₁₁ =v11+v31+v51+v61+z21+z33+z34+z43+z44+z37

h11+h03+h21+h22+h14+h23+h24+k6₁₂=v12+v32+v52+v62+z22+z34+z34+z35+z44+z38

h01+h11+h12+h04+h22+h23+h24+k6₁₃ =v13+v33+v53+z31+v63+z23+z41+z35+z36

h01+h12+h21+h22+h24+k6₁₄=v14+v34+v54+z32+v64+z24+z33+z42+z34+z43+z44+z36

h02+h03+h12+h21+h04+h13+h22+h14+h33+h34+k6₁₅=v15+v35+v55+z33+v65+z25+z25+z36+z46

h03+h04+h13+h22+h14+h23+h34+k6₁₆ =v16+z31+v36+v56+z34+v66+z26+z37+z47

h21+h04+h31+h14+h23+h24+k6₁₇=v17+z31+z32+v37+v57+z35+v67+z27+z45+z38+z48

h01+h02+h11+h03+h12+h21+h04+h13+h14+h32+h24+h33+h34+k6₁₈=v18+z32+v38+z35+v58+z45+v68+z28

h01+h02+h11+h03+h12+h04+h14+h23+k6₂₁ =v61+z32+z33+z42+z43+z35+z36+z38

h02+h11+h03+h12+h21+h04+h13+h24+k6₂₂=v62+z31+z41+z33+z34+z43+z35+z44+z36+z37

h11+h03+h12+h21+h04+h13+h22+h14+k6₂₃=v63+z32+z42+z34=v35+z44+z36+z37+z38

h01+h02+h11+h03+h13+h22+z31+z32+z41+k6₂₄ =v64+z42+z35+z37

h02+h12+h22+h23+h33+z31+k6₂₅ =v05+z34+z36+z37+z46+z47

h03+h21+h13+h31+h23+h24+h34+z32+z34+k6₂₈ =v66+z35+z45+z37+z38+z47+z48

h01+h11+h04+h22+h31+h14+h32+h24+z31+z33+k6₂₇ =v67+z36+z46+z38+z48

h01+h11+h21+h22+h32+z32+z33+z35+z36+z45+k6₂₈ =v68+z46

h12+h04+h13+h22+h14+h23+h34+k6₃₁ =v61+z42+z43+z45+z46+z48

h01+h21+h13+h31+h14+h23+h24+k6₃₂ =v62+z41+z43+z44+z45+z46+z47

h02+h22+h14+h32+h24+k6₃₈ =v63+z42+z44+z45+z46+z47+z48

h11+h03+h12+h21+h04+h13+h22+h14+h33+h34+z41+k6₃₄ =v64+z42+z45+z47

h12+h21+h22+h31+h14+h34+z41+k6₃₅ =v65+z43+z44+z46+z47

h11+h13+h22 30 h31+h23+h32+z42+k6₃₆ =v66+z44+z45+z47+z48

h11+h12+h21+h14+h23+h32+h24+h33+z43+k6₃₇ =v67+z46+z48

h11+h21+h13+h14+h24+h33+z42+z43+z45+k6₃₈ =v68+z46

h12+h04+h13+h22+h14+h23+h34+k6₄₁ =v01+v21+v41+z11+z42+z43+z45+z46+z48

h01+h21+h13+h31+h14+h23+h24+k6₄₂=v02+v22+v42+z12+z41+z43+z44+z45+z46+z47

h02+h22+h14+h32+h24+k6₄₃ =v03+v23+v43+z13+z42+z44+z45+z46+z47+z48

h11+h03+h12+h21+h04+h13+h22+h14+h33+h34+k6₄₄=v04+v24+v44+z14+z41+z42+z45+z47+v37+v57+z35+v67+z27+z45+z38+z48

h12+h21+h22+h31+h14+h34+k6₄₅ =v05+v25+v45+z41+z15+z43+z44+z46+z47

h11+h13+h22+h31+h23+h32+k6₄₆ =v06+v26+v46+z42+z16+z44+z45+z47+z48

h11+h12+h21+h14+h23+h32+h24+h33+k6₄₇ =v07+v27+z41+z47+z43+z17+z46+z48

h11+h21+h13+h14+h24+h33+k6₄₈ =v08+v28+z42+z43=v48+z18+z45+z46

h01+h11+h03+h12+h13+h23+h33+h34+k7₁₁=v01+v21+v41+z11+v71+z31+z32+z34+z35+z45+z37+z38+z47+z48

h01+h02+h11+h12+h21+h04+h13+h14+h24+h34+k7₁₂=v02+v22+v42+z12+z31+v72+z32+z33+z36+z46+z38+z48

h01+h02+h03+h12+h21+h13+h22+h31+h14=k7₁₃=v03+v23v43++z13+z31+z32+v73+z33+z34+z35+z45+z37+z47

h02+h11+h12+h04+h22+h14+h32+h33+h34+k7₁₄=v04+v24+v44+z31+z14+z33+v74+z36+z37+z46+z47

h01+h11+h03+h12+h21+h04+h31+h33+h34+k7₁₅=v05+v25+z31+v45+z32+z41+z15+z42+z34+v75+z35+z44+z37+z38

h02+h12+h04+h13+h22+h32+h34+k7₁₆=v06+v26+z31+z32+z41+v46+z33+z42+z16+z43+v76+z36+z38

h01+h11+h03+h13+h31+h14+h23+h33+k7₁₇=v07+z31+v27+z32+z41+z33+z42+v47+z34+z43+z17+z35+z44+v77+z37

h02+h11+h03+h14+h32+h24+h33+k7₃₈=v08+z31+z41+v28+z33+z43+v48+z18+z36+z37+v48

h01+h21+h13+h32+h33+k7₂₁ =v71+z32+z37+z38+z47+z48

h02+h11+h22+h31+h14+h33+h34+k7₂₂ =v72+z33+z38+z48

h12+h03+h12+h23+h32+h34+z31+k7₂₃ =v73+z34+z35+z45

h12+h04+h31 30 h32+h24+z31+k7₂₄ =v74+z36+z37+z46+z38+z47+z48

h01+h02+h03+h12+h21+h13+h22+h31+h32+h33+z33+z34+z43+k7₂₅=v75+z44+z36+z37

h01+h02+h11+h03+h04+h13+h22+h31+h14+h23+h32+h33+h34+z34+z35+z44+k7₃₅=v76+z36+z37+z38

h02+h03+h12+h21+h04+h14+h23+h32+h24+h33+h34+z31+z41+z36+k7₂₇=v77+z37+z38

h01+h02+h11+h12+h21+h04+h31+h32+h24+h34+z32+z33+z42+z34+z43+z35+z44+z36+k7₂₈=v78+z38

h12+h04+h13+h32+h33+k7₃₁ =v71+z33+z42+z34+z43+z44+z36+z37+z38+z47+z48

h01+h11+h13+h31+h14+h33+h34+k7₃₂ =v72+z34+v43+z44+z37+z38+z48

h02+h12+h14+h32+h34+z31+k7₃₃ =v73+z44+z45+z38

h11+h03+h12+h04+h31+h32+z32+z41+z33+z42+k7₃₄=v74+z34+z43+z35+z44+z36+z37+z46+z38+z47+z48

h01+h11+h12+h32+h24+h34+z31+z32+z34+z43+k7₃₅=v75+z44+z45+z37+z46+z38+z48

h02+h12+h21+h13+h31+h33+z31+z32+z33+z44+k7₃₆ =v76+z45+z46+z38=z47

h11+h03+h13+h22+h31+h14+h32+h34+z31+z32+z41+z33+z34+z35+z45+k7₃₇=v77+z47+z48

h11+h04+h31+h14+h23+h24+h33+h34+z31+z33+z42+z43+z44+z36+z45+z37+k7₃₈=v78+z38+z47

h12+h04+h13+h32+h33+k7₄₁=v11+v31v51++z61+z21+z33+v42+z34+z43+z44+z36+z37+z38+z47+z48

h01+h11+h13+h31+h14+h33+h34+k7₄₂+v12+v32+v52+v62+z22+z34+z43+z44+z37+z38+z48

h02+h12+h14+h32+h34+k7₄₅ =v13+v33+v53+z31+v63+z23+z44+z45+z38

h11+h03+h12+h04+h31+h32+k7₄₄=v14+v34+v54+z32+z41+v64+z24+z33+z42+z34+z43+z35+z44+z36+z37+z46+z38+z47+z48

h01+h11+h12+h32+h24+h34+k7₄₅=v15+v35+z31+z32+v55+v65+z25+z34+z43+v44+z45+z37+z46+v38+z48

h02+h12+h21+h32+h13+h33+k7₄₆=v16+z31+v36+z32+z33+v56+v66+z26+z44+z45+z46+z38+v47

h11+h03+h13+h22+h31+h14+h32+h34+k7₄₇=v17+z31+z32+z41+v37+z33+z34+v57+z35+v67+z27+z45+v46+z47+z48

h11+h04+h31+h14+h23+h24+h33+h34+z31+k7₄₈=v18+z33+z42+v88+z43+z44+v58+z36+z45+v68+z28+z37+z38+z47

h02+h03+h04+h13+h22+h31+h32+h33+k8₁₁=v11+v31+v51+v61+z21+v81+z32+z42+z35+z37

h11+h03+h04+h31+h14+h23+h32+h33+h34+k8₁₂=v12+v32+v52+v62+z22+v82+z33+z43+z35+z36+z38

h11+h12+h21+h04+h32+h24+h33+h34+k8₁₃=v13+v33+v53+z31+v63+z23+z41+v83+z34+z35+z44+z36+z37

h01+h02+h03+h12+h21+h04+h31+h32+h34+k8₁₄=v14+v34+z31+v54+z41+v64+z24+v84+z36+z38

h01+h02+h11+h03+h22+h31+h32+h24+h34+k8₁₅=v15+v35+z33+v35+z33+v65+z25+z35+v85+z36+z45+z37+z46+z47

h01+h02+h03+h12+h21+h04+h31+h23+h32+h33+k8₁₆=v16+z31+v36+z32+v56+z34+v66+z26+z35+v36+z45+v86+z37+z46+z38+z47+z48

h02+h03+h21+h04+h13+h22+h31+h32+h24+h33+h34+k8₁₇=v17+z31+z32+v37+z33+v57+v67+z27z36+z37+z46+v87+z38+z47+z48

h01+h02+h21+h04+h31+h14+h23+h24+h33+k8₁₅=v18+z32+v38+z34+z35+v58+z36+z45+v68+z28+z46+z38+v88+z48

h01+h02+h11+h21+h23+h33+z31+k8₂₁ =v81+z41+z33+z43+z35

h02+h03+h12+h21+h22+h31+h24+h34+z31+z32+z41+k8₂₂ =v82+z42+z34+z44=z36

h01+h03+h21+h04+h13+h22+h31+h23+h32+z31+z32+z41+z33+z42+k8₂₃=v83+z43+z37

h01+h04+h22+h14+h32+h24+z32+z42+z34+k8₂₄ =v84+z44+z38

h11+h12+h21+h04+h22+h31+h32+h24+h33+h34+z32+z33+z34+k8₂₅ =v85+z28+z48

h01+h12+h13+h22+h23+h32+h32+h24+h33+h34+z33+z34+z35+z45+k8₂₆ +v86

h02+h11+h13+h14+h23+h24+h33+h34+z34+z36+z46+k8₄₅ =v87

h11+h03+h21+h04+h22+h31+h14+h23+h32+h33+z31+z32+z33+z34+z37+z38+z47+k8₂₃=v88+z48

h01+h21+h04+h22+h23+h24+h34+k8₃₁=v81+z41+z43+z35+z36+v65+z25+z37+z46+v47

h01+h02+h22+h31+h23+h24+h33+z02+k8₃₁=v82+z42+z35+z44+z36+z45+z37+z38+z47+z48

h02+h03+h31+h23+h32+h24+h34+z41+z42+k8₃₃ =v83+z43+z36+z37+z46+z38+z48

h03+h21+h22+h31+h23+h33+h34+z42+k8₃₄ =v84+z35+z44+z36+z45+z46+z38

h22+h14+h24+h33+z32+z34+z34+z43+k8₃₈ =v85+z48

h11+h21+h31+h23+h34+z31+z41+z33+z44+z45+k8₃₀ =v86

h12+h21+h22+h31+h32+h24+z31+z32+z41+z42+z46+k8₃₇ =v87

h21+h13+h14+h23+h32+h24+z31+z33+z42+z34+z47+k8₃₆ =v88+z48

h01+h21+h04+h22+h23+h24+h34+k8₄₁=v01+v21+v41+z11+v71+z41+z43+z35+z36+z37+z46+z47

h01+h02+h22+h31+h23+h24+h33+k8₄₂=v02+v22+v42+z12+v72+z42+z35+z44+z36+z45+z37+z38+z47+z48

h02+h03+h31+h23+h32+h24+h34+k8₄₃=v03+v23+v43+z13+z41+v73+z42+z43+v73+z42+z43+z36+z37+z46+z38+z48

h03+h21+h22+h31+h23+h33+h34+k8₄₄=v04+v24+v44+z14+z42+v74+z35+z44z36+z45+z46+z38

h22+h14+h24+h33+k8₄₅ =v05+v25+v45+z32+z15+z34+z43+v75+z48

h11+h21+h31+h23+h34+k8₄₆ =v06+v26+z31+z41+v46+z23+z16+z44+v76+v45

h12+h21+h22+h31+h32+h24+k8₄₇=v07+z31+v27+z32=z41+z42+v47+z34+z17=v77+z46

h21+h13+h14+h23+h32+h24+k8₄₃=v08+z31+v28+z33+z42+z34+v48=z18+v78+z47+z48

h02+h12+h21+h31+h23+k9₁₁=v01+v21+v71+z31+z32+v01+z34=z36+z37+z46+z38+z47+z48

h03+h21+h13+h22+h32+h24+k9₁₂=v02+v22+v42+z12+z31+v72+z33=v92+z37+z38+z47+z48

h01+h11+h21+h04+h22+h23+h33+k9₁₃+v03+v23+v43+z13+z31=z32+v73+z33+z34+v93+z38+z48

h01+h11+h22+h24+h34+k9₁₄=v04+v24+v44+z31+z14+z33=v74+z35+v94+z36+z45+z37+z46+z28+z47+z48

h01+h02+h12+h04+h31+h14+h32+h33+h34+k9₁₅=v05+v25+z31+z31+v45+z41+z15=v75+z35+v95+z38

h01+h02+h11+h03+h13+h32+h33+h34+k9₁₆=v06+v26+z32+v46+z42+z16+z35+v76+z36+z96

h01+h02+h11+h03+h12+h04+h14+z33+h34+k9₁₇=v07+v27+z33+v47+z43+z17+z36+v77+z37+v97

h01+h02+h11+h03+h13+h31+h14+h32+h33+k9₁₈=v08+v28+z34+v48+z44+z18+z37+v78+v98

h01+h11+h03+h21+h13+z32 30 k9₂₁ =v91+z35+z36+z45+z45

h01+h02+h11+h12+h04+h22+h14+z33+k9₂₂ =v92+z36+z37+z46+v47

h01+h02+h11+h03+h12+h13+h23+z31+k9₂₃ =v03+z25+z45+z37+z38+z47+z48

h02+h12+h04+h14+h24+z31+z35+k9₂₄ =v94+z45+z38+z48

h01+h02+h11+h03+h12+h04+h13+h31+h41+z31+z32+z41+z42+z36+k9₂₅ =v95+z38

h01+h03+h12+h04+h13+h14+h32+z32+z32+z33+z42+z43+z35+z37k9₂₆ =v96

h03+h04+h13+h14+h33+z31+z41+z33+z34+z43+z35+z44+z36+z38k9₂₇ =v97

h01+h02+h11+h03+h12+h13+h34+z31+z41+z34+z35+z44+z37+z38k9₂₈ =v98[formula 34]

[0133] Then, the next step S104 is executed to carry out amatricial-equation transformation process. In this process, vectors K,H, U and V are set as follows.

K=(k1₁₁, K1₁₂, . . . , k9₂₈)

H=(h01, h02, . . . , h44)

U=(z11, z12, . . . , z44)

V=(v01, v02, . . . , v74)   [formula 35]

[0134] With the vectors K, H, U and V set as expressed by the aboveequations, the simultaneous linear equation can be transformed into thefollowing matricial equation.

[0135] [Formula 36] ${M_{KH}\begin{pmatrix}{\,^{t}K} \\{\,^{t}H}\end{pmatrix}} = {M_{UV}\begin{pmatrix}{\,^{t}U} \\{\,^{t}V}\end{pmatrix}}$

[0136] It is to be noted that, in the above equation, symbols M_(KH) andM_(UV) each denote a GF(2) matrix comprising coefficients of thesimultaneous linear equation described above.

[0137] Then, the next step S105 is executed to carry out a unitarytransformation process.

[0138] Let symbol N_(r) denote the rank value of the matrix M_(UV) asfollows:

[0139] rank(M_(UV))=N _(r)   [formula 37]

[0140] Then, let symbol Nm denote the number of rows composing thematrix M_(UV). By multiplying both the left-hand and right-hand sides ofthe matricial equation by a row-deform unitary matrix Q from the left,the matrix M_(UV) can be deformed into a step matrix. In this process, asmall matrix consisting of (N_(m)-N_(r)) lowest rows of the matrixQM_(UV) becomes a null matrix.

[0141] Then, the next step S106 is executed to carry out a small-matrixselection process. Let symbol M*_(KH) denote a small matrix consistingof (N_(m)-N_(r)) lowest rows of the matrix QM_(KH). In this case, thesmall matrix M*_(KH) becomes a null matrix (O) as expressed by thefollowing equation.

M*_(KH)=0  [formula 38]

[0142] Then, the next step S107 is executed to carry out alinear-relation equation generation process. This matricial equation istransformed into linear-relation equations, which are each associatedwith a row. Then, actual values are substituted for h01, h02, . . . andh44 to obtain the following relation equations:

[0143] [Formula 39]

0×07=k1₁₁ +k1₂₁ +k1₂₄ +k ₂₆ +k1₃₁ +k1₃₄ +k1₃₆ +k2₄₂ +k2₁₂ +k2₂₃ +k3₁₁+k3₂₁

0×66=k1₁₂ +k1₂₁ +k1₂₂ +k1₂₇ +k1₃₁ +k1₃₂ +k1₃₇ +k1₄₃ +k2₁₃ +k2₂₃ +k3₁₂+k3₂₂

0×9e=k1₁₂ +k1₂₂ +k1₂₃ +k1₂₅ +k1₂₈ +k1₃₂ +k1₃₃ +k1₃₅ +k1₃₈ +k1₄₁ +k1₄₄+k2₁₁ +k2₁₄ +k2₂₄ +k3₁₃ +k3₂₃

0×df=k1₁₄ +k1₂₃ +k1₂₅ +k1₃₃ +k1₃₅ +k1₄₁ +k2₁₁ +k2₂₁ +k3₁₄ +k3₂₄

0×e9=k1₁₅ +k1₂₂ +k1₂₄ +k1₂₅ +k1₂₆ +k1₃₂ +k1₃₄ +k1₃₅ +k1₃₈ +k1₄₆ +k1₄₈+k2₁₆ +k2₁₈ +k2₂₆ +k2₂₈ +k3₁₆ +k3₂₅

0×23=k1₁₈ +k1₂₁ +k1₂₃ +k1₂₆ +k1₂₇ +k1₃₁ +k1₃₃ +k1₃₆ +k1₃₇ +k1₄₅ +k1₄₇+k2₁₅ +k2₁₇ +k2₂₅ +k2₂₇ +k3₁₈ +k3₂₈

0×60=k1₁₇ +k1₂₁ +k1₂₂ +k1₂₄ +k1₂₅ +k1₂₇ +k1₂₈ +k1₃₁ +k1₃₂ +k1₃₄ +k1₃₅+k1₃₇ +k1₃₈ +k1₄₅ +k1₄₀ +k1₄₈ +k2₁₅ +k2₁₆ +k2₁₅ +k2₁₆ +k2₁₈ +k2₂₅ +k2₂₆+k2₂₈ +k3₁₇ +k3₃₇

0×cd=k1₁₈ +k1₂₂ +k1₂₃ +k1₂₄ +k1₂₅ +k1₂₈ +k1₃₁ +k1₃₃ +k1₃₄ +k1₃₅ +k1₃₈+k1₄₅ +k1₄₇ +k1₄₈ +k2₄₈ +k2₁₅ +k2₁₇ +k2₁₈ +k2₂₃ +k2₂₇ +k2₂₈ +k3₃₈ +k3₂₈

0×3d=k1₂₁ +k1₂₄ +k1₂₇ +k1₂₈ +k1₃₁ +k1₃₄ +k1₃₇ +k1₃₈ +k1₄₁ +k1₄₂ +k1₄₃+k2₁₁ +k2₄₈ +k2₁₁ +k2₁₂ +k2₁₈ +k2₂₁ +k2₂₂ +k2₂₃ +k2₁₃ +k4₁₃ +k3₃₃ ₃₃

0×00=k1₂₂ +k1₃₅ +k1₂₆ +k1₂₇ +k1₂₈ +k1₃₂ +k1₃₅ +k1₃₆ +k1₃₇ +k1₃₈ +k1₄₁+k1₄₃ +k1₄₆ +k1₄₇ +k1₁₈ +k2₁₃ +k2₁₆ +k2₁₇ +k2₁₈ +k2₂₁ +k2₂₃ +k2₂₆ +k2₂₇+k2₂₈ +k4₁₁ +k4₃₁

0×e1=k1₂₃ +k1₂₆ +k1₂₇ +k1₂₅ +k1₃₃ +k1₃₅ +k1₃₇ +k1₃₈ +k1₄₁ +k1₄₂ +k1₄₄+k1₄₇ +k1₄₈ +k2₁₁ +k2₁₄ +k2₁₇ +k2₁₆ +k2₂₁ +k2₂₂ +k2₂₄ +k2₂₇ +k2₂₈ +k4₁₂+k4₃₂

0×80=k1₂₄ +k1₂₅ +k1₂₆ +k1₂₈ +k1₃₄ +k1₃₅ +k1₃₆ +k1₃₈ +k1₄₁ +k1₄₃ +k1₄₄+k1₄₅ +k1₄₆ +k1₄₇ +k2₁₁ +k2₁₅ +k2₁₆ +k2₁₇ +k2₂₁ +k2₂₃ +k2₂₄ +k2₂₅ +k2₂₆+k2₂₇ +k4₃₂ +k4₁₆ +k4₃₃ +k4₃₄

0×39=k1₂₅ +k1₃₅ +k1₄₃ +k1₄₄ +k1₄₅ +k1₄₇ +k1₄₅ +k2₁₁ +k2₁₂ +k2₁₄ +k2₁₅+k2₁₆ +k2₁₇ +k2₁₈ +k2₂₁ +k2_(24i +k)2₂₅ +k2₂₇ +k2₂₈ +k4₁₂ +k4₁₆ +k4₃₂+k4₃₈

0×2d=k2₂₆ +k1₃₆ +k1₄₁ +k1₄₂ +k1₄₆ +k1₄₈ +k2₁₁ +k2₁₂ +k2₁₃ +k2₁₆ +k2₁₇+k2₁₈ +k2₁₁ +k2₂₂ +k2₂₆ +k2₂₈ +k4₁₃ +k4₁₇ +k4₃₃ +k4₃₇

0×d5d=k1₂₇₆ +k1₂₈ +k1₃₇ +k1₃₈ +k1₄₂ +k1₄₅ +k1₄₈ +k2₁₄ +k2₁₅ +k2₁₆ +k2₁₈+k2₂₂ +k2₂₅ +k2₂₆ +k4₁₄ +k4₁₈ +k4₃₄ +k4₃₈

0×fα=k1₂₈ +k1₃₆ +k1₄₃ +k1₄₆ +k1₄₇ +k2₁₁ +k2₁₃ +k2₁₅ +k2₁₆ +k2₁₇ +k2₂₃+k2₂₆ +k2₂₇ +k4₁₁ +k4₁₅ +k4₃₁ +k4₃₅

0×39=k1₄₁ +k1₄₂ +k1₄₄ +k1₄₆ +k1₄₈ +k2₁₃ +k2₁₇ +k2₁₈ +k2₂₂ +k2₂₄ +k2₂₅+k2₂₈ +k2₃₂ +k4₁₁ +k4₁₃ +k4₁₃ +k4₂₄ +k4₁₈ +k4₁₇ +k4₃₃ +k4₃₃ +k4₃₄ +k4₃₆+k4₃₇

0×35=k1₄₂ +k1₄₃ +k1₄₄ +k1₄₄ +k1₄₅ +k1₄₆ +k1₄₇ +k2₁₁ +k2₁₃ +k2₁₆ +k2₂₁+k2₂₂ +k2₂₃ +k2₂₅ +k26+k2₂₇ +k2₃₁ +k2₃₅ +k4₄₁ +k4₁₁ +k4₁₂ +k4₁₅ +k4₁₇+k4 30 k4₃₁ +k4₃₂ +k3₃₅ +k4₃₇

0×4b=k1₄₃ +k1₄₄ +k1₃₅ +k1₄₆ +k1₄₇ +k2₁₁ +k2₁₂ +k2₁₃ +k2₁₄ +k2₁₇ +k2₂₁+k2₂₂ +k2₂₃ +k2₂₄ +k2₂₅ +k2₂₇ +k2₂₈ +k2₃₁ +k2₃₂ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₅+k4₁₆ +k4₁₈ +k4₃₁ +k4₃₂ +k4₃₃ +k4₃₆ +k4₃₈

0×e7=k1₄₄ +k1₄₅ +k1₄₇ +k1₄₈ +k2₁₁ +k2₁₂ +k2₁₃ +k2₁₅ +k2₁₈ +k2₂₂ +k2₂₃+k2₂₄ +k2₂₆ +k2₂₇ +k2₂₈ +k2₃₂ +k2₃₃ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₄ +k4₁₅ +k4₁₆+k4₁₇ +k4₃₁ +k4₃₃ +k4₃₄ +k4₃₅ +k4₃₆ +k4₃₇

0×33=k1₄₅ +k1₄₆ +k2₁₂ +k2₁₃ +k3₁₄ +k2₁₆ +k2₂₁ +k2₂₅ +k2₂₆ +k2₃₁ +k3₁₁+k4₁₂ +k4₁₃ +k4₁₄ +k4₁₅ +k4₃₂ +k4₃₃ +k4₃₄ +k4₃₅ +k5₁₁ +k5₂₁

0×db=k1₄₅ +k1₄₇ +k2₁₃ +k2₁₄ +k2₁₇ +k2₂₂ +k2₂₆ +k2₂₇ +k2₃₂ +k3₁₂ +k4₁₃+k4₁₄ +k4₁₆ +k4₃₃ +k4₃₄ +k4₃₆ +k5₁₂ +k5₂₂

0×8f=k1₄₇ +k1₄₈ +k2₁₁ +k2₁₃ +k2₁₄ +k2₁₅ +k21₁₆ +k2₁₇ +k2₂₁ +k2₂₂ +k2₂₄+k2₂₇ +k2₂₈ +k2₃₁ +k2₃₂ +k2₃₄ +k3₁₁ +k2₁₂ +k3₁₄ +k4₁₁ +k4₁₃ +k4₁₄ +k4₁₅+k4₁₆ +k4₁₈ +k4₃₁ +k4₃₂ +k4₃₄ +k4₃₅ +k4₃₈ +k5₁₁ +k5₁₂ +k5₁₄ +k5₂₂ +k5₃₂+k5₂₄

0×83=k1₄₈ +k2₁₂ +k2₁₄ +k2₁₆ +k2₁₇ +k2₁₈ +k2₂₁ +k2₂₂ +k1₂₃ +k2₂₈ +k2₃₁+k2₃₂ +k2₃₃ +k3₁₁ +k3₁₂ +k4₁₃ +k4₁₄ +k4₁₅ +k4₁₆ +k4₁₇ +k4₃₂ +k4₃₄ +k4₃₅+k4₃₆ +k4₃₇ +k5₁₁ +k5₁₂ +k5₁₃ +k5₂₁ +k5₂₂ +k5₂₃

0×00=k2₁₁ +k3₁₁ +k4₁₁ +k4₃₁ +k4₄₁

0×00=k2₁₂ +k3₁₂ +k4₁₂ +k4₃₂ +k4₄₂

0×00=k2₁₃ +k3₁₃ +k4₁₃ +k4₃₃ +k4₄₃

0×00=k2₁₄ +k3₁₄ +k4₁₄ +k4₂₄ +k4₄₄

0×1f=k2₁₅ +k2_(17i +k)2₂₂ +k2₂₃ +k2₂₄ +k2₂₅ +k2₃₂ +k2₃₃ +k2₂₄ +k2₃₅+k3₁₁ +k3₁₂ +k3₁₃ +k3₁₄ +k4₁₅ +k5₁₇ +k4₃₅ +k4₃₇ +k4₄₁ +k4₄₂ +k4₄₃ +k4₄₄

0×8e=k2₁₆ +k2₁₇ +k2₁₈ +k2₂₂ +k2₂₅ +k2₂₆ +k2₃₂ +k2₃₅ +k2₃₆ +k3₁₁ +k4₁₆+k4₁₇ +k4₁₈ +k4₃₆ +k4₃₇ +k4₃₆ +k4₃₇ +k4₃₈ +k4₄₁

0×68=k2₁₇ +k2₁₅ +k2₂₃ +k2₂₅ +k2₂₇ +k2₃₃ +k2₃₇ +k3₁₂ +k4₁₇ +k4₁₈ +k4₃₇+k4₃₈ +k4₄₂

0×35d=k2₁₈ +k2₂₁ +k2₂₄ +k2₂₅ +k2₂₇ +k2₂₈ +k2₃₁ +k2₃₄ +k2₃₆ +k2₃₇ +k2₃₈+k3₁₃ +k4₁₈ +k4₃₈ +k4₄₂

0×42=k2₂₂ +k2₂₃ +k2₂₆ +k2₂₈ +k2₂₃ +k2₃₂ +k2₃₆ +k2₃₈ +k3₁₁ +k3₁₄ +k3₁₅+k4₄₁ +k4₄₄ +k4₄₅

0×e6=k2₂₂ +k2₂₃ +k2₂₅ +k2₂₇ +k2₃₂ +k2₃₃ +k2₂₅ +k2₃₇ +k3₁₁ +k3₁₂ +k3₁₆+k4₄₁ +k4₃₂ +k4₄₆

0×91=k2₂₃ +k2₂₄ +k2₂₈ +k2₃₃ +k2₃₄ +k2₃₆ +k3₁₂ +k3₁₃ +k3₁₄ +k3₁₅ +k3₁₈+k4₄₂ +k4₄₃ +k4₄₄ +k4₄₅ +k4₄₆ +k4₄₈

0×f9=k2₂₄ +k2₂₇ +k2₃₄ +k2₃₇ +k3₁₃ +k3₁₄ +k3₁₅ +k3₁₆ +k3₁₇ +k4₄₃ +k4₄₄+k4₄₅ +k4₄₆ +k4₄₇

0×α0=k2₂₅ +k2₂₈ +k2₃₅ +k2₃₈ +k3₁₁ +k3₁₃ +k3₁₅ +k5₁₁ +k5₁₃ +k5₁₄ +k5₁₅+k5₂₁ +k5₂₃ +k5₂₄ +k5₂₅

0×v7=k2₂₆ +k2₂₈ +k2₃₈ +k2₃₆ +k2₂₃ +k3₁₂ +k3₁₃ +k3₁₆ +k4₄₁ +k4₄₄ +k5₁₁+k5₁₂ +k5₁₃ +k5₁₅ +k5₁₆ +k5₂₁ +k5₂₂ +k5₂₃ +k5₂₅ +k5₂₆

0×07=k2₂₇ +k2₂₈ +k2₃₇ +k2₃₈ +k3₁₁ +k3₁₄ +k3₁₅ +k3₁₆ +k3₁₇ +k4₄₁ +k4₄₂+k4₄₄ +k5₁₂ +k5₁₅ +k5₁₆ +k5₁₇ +k5₂₂ +k5₂₅ +k5₂₆ +k5₂₇

0×c1=k2₂₈ +k2₃₈ +k3₁₁ +k3₁₂ +k3₁₅ +k3₁₆ +k3₁₇ +k3₁₈ +k4₄₁ +k4₄₂ +k4₄₃+k5₁₈ +k5₁₅ +k5₁₆ +k5₁₇ +k5₁₈ +k5₂₂ +k5₂₅ +k5₂₆ +k5₂₇ +k5₂₈

0×c9=k2₄₁ +k3₁₁ +k3₁₂ +k3₁₃ +k3₁₆ +k3₁₇ +k3₂₁ +k4₄₁ +k4₄₂ +k4₄₃ +k4₄₇+k4₁₈ +k5₁₁ +k5₁₆ +k5₁₈ +k5₂₁ +k5₁₆ +k5₂₆ +k5₂₈

0×ed=k2₄₂ +k3₁₁ +k3₁₂ +k3₁₃ +k3₁₄ +k3₁₆ +k3₁₇ +k3₁₈ +k3₂₂ +k4₄₁ +k4₄₂+k4₄₃ +k4₄₄ +k4₄₈ +k5₁₂ +k5₁₅ +k5₁₇ +k5₂₂ +k5₂₅ +k5₂₇

0×f6=k2₄₅ +k3₁₂ +k3₁₃ +k3₁₄ +k3₁₆ +k3₁₅ +k3₂₃ +k4₄₂ +k4₄₃ +k4₄₄ +k4₄₅+k5₁₂ +k5₁₅ +k5₁₆ +k5₁₈ +k5₂₃ +k5₂₅ +k5₂₃ +k5₂₈

0×46=k2₄₄ +k3₁₁ +k3₁₂ +k3₁₄ +k3₁₅ +k3₁₆ +k3₂₄ +k4₄₁ +k4₄₂ +k4₄₄ +k4₄₅+k4₄₇ +k4₄₈ +k5₁₅ +k5₁₇ +k5₁₈ +k5₂₄ +k5₂₆ +k5₂₇ +k5₂₈

0×81=k2₄₅ +k3₁₁ +k3₁₄ +k3₁₅ +k3₁₆ +k3₁₇ +k3₁₈ +k3₂₅ +k4₄₁ +k4₄₃ +k4₄₆+k4₄₈ +k5₁₂ +k5₁₄ +k5₂₃ +k5₂₄ +k5₂₇

0×29=k2₄₆ +k3₁₁ +k3₁₂ +k3₁₆ +k3₁₇ +k3₁₈ +k3₂₆ +k4₄₁ +k4₁₂ +k4₄₄ +k4₄₅+k4₄₇ +k5₁₄ +k5₁₅ +k5₂₄ +k5₂₅ +k5₂₈

0×85=k2₄₇ +k3₁₂ +k3₁₃ +k3₁₇ +k3₁₈ +k3₂₇ +k4₄₁ +k4₄₂ +k4₄₃ +k4₄₅ +k4₄₆+k4₄₈ +k5₁₁ +k5₁₅ +k5₁₆ +k5₂₁ +k5₂₅ +k5₂₆

0×69=k2₄₈ +k3₁₃ +k3₁₅ +k3₁₆ +k3₁₇ +k3₂₈ +k4₄₂ +k4₄₄ +k4₄₅ +k4₄₇ +k4₄₈+k5₁₂ +k5₁₃ +k5₁₄ +k5₁₆ +k5₂₂ +k5₂₃ +k4₂₄ +k5₂₈

0×8b=k3₁₁ +k3₁₂ +k3₁₆ +k3₁₇ +k3₁₈ +k3₂₁ +k4₄₁ +k4₄₃ +k5₁₆ +k5₁₇ +k5₁₈+k5₂₆ +k5₂₇ +k5₂₈ +k5₄₁

0×1b=k3₁₂ +k3₁₃ +k3₁₄ +k3₁₆ +k4₁₁ +k4₁₂ +k4₄₂ +k4₄₃ +k4₄₄ +k5₁₅ +k5₂₆+k5₄₁ +k5₄₂

0×bc=k3₁₃ +k3₁₄ +k3₁₇ +k4₁₂ +k4₁₃ +k4₄₃ +k4₄₄ +k4₁₇ +k5₂₇ +k5₄₂ +k5₄₃

0×53=k3₁₄ +k3₁₅ +k3₁₈ +k4₁₁ +k4₁₄ +k4₄₄ +k5₁₅ +k5₁₈ +k5₂₅ +k5₂₉ +k5₄₂+k5₄₃ +k5₄₄

0×04=k3₁₅ +k3₁₇ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₅ +k4₄₁ +k4₄₄ +k4₄₅ +k4₄₆ +k4₄₇+k4₄₈ +k5₁₁ +k5₁₄ +k5₁₆ +k5₁₈ +k5₂₁ +k5₂₄ +k5₂₆ +k5₂₈ +k5₄₁ +k5₄₂ +k5₄₃+k5₄₅

0×cf=k3₁₆ +k3₁₇ +k3₁₈ +k4₁₄ +k4₁₅ +k4₁₆ +k4₄₂ +k4_(44k)3₁₅ +k3₁₇ +k4₁₁+k4₁₂ +k4₁₃ +k4₁₅ +k4₄₁ +k4₄₄ +k4₄₅ +k4₄₆ +k5₁₂ +k5₁₄ +k5₁₅ +k5₁₆ +k5₁₇+k5₁₈ +k5₂₂ +k5₂₄ +k5₂₅ +k5₂₈ +k5₂₇ +k5₄₄ +k5₄₅ +k5₄₈

0×58=k3₁₇ +k3₁₆ +k4₁₁ +k4₁₆ +k4₁₇ +k4₄₁ +k4₄₂ +k4₄₅ +k5₁₁ +k5₁₂ +k5₁₅+k5₁₇ +k5₁₈ +k5₂₁ +k5₂₃ +k5₄₁ +k5₄₆ +k5₄₇

0×21=k3₁₇ +k4₁₂ +k4₁₅ +k4₁₇ +k4₁₈ +k4₄₁ +k4₄₂ +k4₄₇ +k5₁₁ +k5₁₂ +k5₁₄+k5₁₇ +k5₁₈ +k5₂₁ +k5₂₂ +k5₂₄ +k5₂₇ +k5₂₈ +k5₄₂ +k5₃₄ +k5₄₇ +k5₄₈

0×37=k3₂₁ +k3₃₁ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₄ +k4₁₆ +k4₁₇ +k4₁₈ +k4₁₁ +k4₄₄+k4₄₄ 450 k4₄₇ +k5₁₁ +k5₁₄ +k5₁₅ +k5₁₇ +k5₂₁ +k5₂₄ +k5₂₅ +k5₂₇ +k5₄₁+k5₄₂ +k5₄₃ +k5₄₄ +k5₄₆ +k5₄₆ +k5₄₈

0×α3=k3₂₂ +k3₃₂ +k4₁₂ +k4₁₃ +k4₁₄ +k4₁₇ +k4₁₈ +k4₄₁ +k4₄₂ +k4₄₈ +k5₁₁+k5₁₂ +k5₁₅ +k5₁₆ +k5₂₁ +k5₂₂ +k5₂₅ +k5₂₆ +k5₂₈ +k5₄₂ +k5₄₃ +k5₄₄ ′k5₄₇+k5₄₈

0×9b=k3₂₈ +k3₃₃ +k4₁₄ +k4₁₈ +k4₄₂ +k4₄₃ +k4₄₅ +k4₄₇ +k5₁₂ +k5₁₃ +k5₁₅+k5₁₆ +k5₁₇ +k5₂₃ +k5₂₆ +k5₂₈ +k5₂₇ +k5₄₃ +k5₄₄ +k5₄₈

0×51=k3₂₄ +k3₃₄ +k4₁₁ +k4₁₂ +k4₁₃ +k4₁₅ +k4₁₆ +k4₁₇ +k4₁₈ +k4₄₃ +k4₄₆+k4₄₈ +k5₁₃ +k5₁₆ +k5₂₃ +k5₂₆ +k5₄₁ +k5₄₂ +k5₄₃ +k5₄₅ +k5₄₆ +k5₄₇ +k5₄₈

0×4α=k3₂₅ +k3₃₈ +k4₁₂ +k4₁₄ +k4₁₆ +k4₁₇ +k4₁₈ +k4₄₁ +k4₄₆ +k4₄₇ +k5₁₁+k5₁₆ +k5₁₇ +k5₂₁ +k5₂₆ +k5₁₇ +k5₂₁ +k5₂₆ +k5₂₇ +k5₄₂ +k5₄₄ +k5₄₅ +k5₄₇+k5₄₈

0×4f=k3₂₆ +k3₃₆ +k4₁₁ +k4₁₃ +k4₁₇ +k4₁₈ +k4₄₂ +k4₄₅ +k4₄₇ +k4₄₃ +k5₁₂+k5₁₅ +k5₁₇ +k5₁₈ +k5₂₂ +k5₂₅ +k5₂₇ +k5₂₈ +k5₄₁ +k5₄₃ +k5₄₇ +k5₄₈

0×8α=k3₂₇ +k3₃₇ +k4₁₁ +k4₁₂ +k4₁₄ +k4₁₈ +k4₄₃ +k4₄₆ +k4₄₈ +k5₁₃ +k5₁₆+k5₁₈ +k5₂₃ +k5₃₆ +k5₂₈ +k5₄₁ +k5₄₂ +k5₄₄ +k5₄₈

0×c2=k3₂₈ +k3₃₈ +k4₁₁ +k4₁₃ +k4₁₄ +k4₁₅ +k4₁₆ +k4₁₇ +k4₁₈ +k4₄₄ +k4₄₅+k4₄₆ +k5₁₄ +k5₁₅ +k5₁₆ +k5₂₄ +k5₂₅ +k5₂₆ +k5₄₁ +k5₄₂ +k5₄₄ +k5₄₅ +k5₄₇+k5₄₈

0×72=k3₄₁ +k4₁₄ +k4₁₅ +k4₃₁ +k4₄₃ +k4₄₇ +k5₁₃ +k5₁₇ +k5₂₃ +k5₂₇ +k5₄₁+k5₄₄ +k5₄₈

0×α2=k3₁₂ +k4₁₁ +k4₁₅ +k4₃₂ +k4₄₁ +k4₄₄ +k4₄₅ +k5₁₁ +k5₁₄ +k5₁₅ +k5₁₈+k5₂₁ +k5₂₄ +k5₂₆ +k5₂₈ +k5₄₁ +k5₄₃ +k5₄₅

0×68=k3₁₃ +k4₁₂ +k4₁₆ +k4₃₃ +k4₄₁ +k4₄₂ +k4₄₅ +k5₁₁ +k5₁₂ +k5₁₆ +k5₂₁+k5₂₂ +k5₂₅ +k5₂₆ +k5₂₈ k5₄₂ +k5₄₃ +k5₄₆

0×56=k3₄₄ +k4₁₃ +k4₁₄ +k4₁₇ +k4₁₈ +k4₂₄ +k4₄₆ +k5₁₂ +k5₁₆ +k5₂₂ +k5₂₈+k5₄₃ +k5₄₇ +k5₄₈

0×80=k3₄₅ +k4₁₂ +k4₁₃ +k4₁₅ +k4₁₈ +k4₃₅ +k4₄₂ +k4₄₃ +k4₄₇ +k5₁₂ +k5₁₃+k5₁₇ +k5₂₂ +k5₂₃ +k5₂₇ +k5₄₂ +k5₄₃ +k5₄₅ +k5₄₈

0×dα=k3₄₆ +k4₁₁ +k4₁₂ +k4₁₄ +k4₁₅ +k4₁₇ +k4₃₆ +k4₁₁ +k4₁₃ +k4₄₄ +k4₄₅+k5₁₁ +k5₁₃ +k5₁₄ +k5₁₅ +k5₁₈ +k5₂₁ +k5_(23i +k)5₂₄ +k5₂₅ +k5₂₈ +k5₄₁+k5₄₄ +k5₄₅ +k5₄₆ +k5₄₈

0×c4=k3₄₇ +k4₁₂ +k4₁₄ +k4₁₅ +k4₁₆ +k4₁₈ +k4₃₇ +k4₄₆ +k5₁₂ +k5₁₄ +k5₁₅+k5₁₆ +k5₂₂ +k5₂₄ +k5₂₅ +k5₂₆ +k5₄₂ +k5₄₄ +k5₄₅ +k5₄₆ +k5_(47i +k)5₄₈

0×2c=k3₄₈ +k4₁₁ +k4₁₂ +k4₁₅ +k4₁₇ +k4₁₈ +k4₃₈ +k4₄₁ +k4₄₂ +k4₄₈ +k5₁₁+k5₁₂ +k5₁₆ +k5₂₁ +k5₂₁ +k5₂₂ +k5₂₆ +k5₄₁ +k5₄₂ +k5₄₅ +k5₄₇

0×5e=k4₁₁ +k4₁₃ +k4₁₄ +k4₁₅ +k4₁₆ +k4₁₇ +k4₁₈ +k4₄₁ +k4₄₄ +k4₄ +k4₄₅+k4₄₈ +k5₁₄ +k5₁₆ +k5₁₈ +k5₂₁ +k5₂₄ +k5₂₆ +k5₂₈ +k5₄₁ +k5₄₃ +k5₄₄ +k5₄₅+k5₄₈ +k5₄₇ +k5₄₈ +k6₁₁ +k6₃₁

0×69=k4₁₂ +k4₁₄ +k4₁₆ +k4₁₇ +k4₁₈ +k4₄₁ +k4₄₂ +k4₄₅ +k4₄₇ +k5₁₁ +k5₁₅+k5₁₇ +k5₂₁ +k5₂₂ +k5₂₅ +k5₂₇ +k5₄₃ +k5₄₄ +k5₄₆ +k5₄₇ +k5₄₈ +k6₁₂ +k6₃₂

0×66=k4₁₃ +k4₁₄ +k4₁₅ +k4₁₈ +k4₄₁ +k4₄₂ +k4₄₃ +k4₄₈ +k5₁₁ +k5₁₃ +k5₁₃+k5₁₄ +k5₁₈ +k5₂₁ +k5₂₂ +k5₂₃ +k5₂₉ +k5₄₃ +k5₄₄ +k5₄₅ +k5₄₈ +k6₁₂ +k6₁₄+k6₃₂ +k6₃₄

0×94=k4₁₄ +k4₁₅ +k4₁₆ +k4₄₃ +k4₄₂ +k4₄₃ +k4₄₄ +k4₄₅ +k5₁₂ +k5₁₄ +k5₁₅+k5₂₁ +k5₂₂ +k5₂₃ +k5₂₄ +k5₂₅ +k5₄₄ +k5₄₅ +k5₄₆ +k6₁₁ +k6₁₃ +k6₃₁ +k6₃₃

0×8f=k4₁₅ +k4₁₆ +k4₄₁ +k4₄₂ +k4₄₃ +k4₄₄ +k5₁₆ +k5₂₁ +k5₂₂ +k5₂₃ +k5₂₄+k5₄₃ +k5₄₆ +k6₁₁ +k6₁₂ +k6₁₃ +k6₁₄ +k6₁₆ +k6₃₁ +k6₃₂ +k6₃₃ +k6₃₄ +k6₃₅

0×1b=k4₁₆ +k4₁₇ +k4₄₂ +k4₄₃ +k4₄₄ +k5₁₆ +k5₂₂ +k5₂₃ +k5₂₄ +k5₄₆ +k5₄₇+k6₁₂ +k6₁₃ +k6₁₄ +k6₁₆ +k6₃₂ +k6₃₃ +k6₃₄ +k6₃₆

0×48=k4₁₇ +k4₁₈ +k4₄₂ +k4₄₃ +k5₁₅ +k5₁₆ +k5₁₈ +k5₂₂ +k5₂₃ +k5₄₇ +k5₄₈+k6₁₂ +k6₁₃ +k6₁₆ +k6₁₈ +k6₃₂ +k6₃₅ +k6₃₆ +k6₃₈

0×28=k4₁₈ +k4₄₂ +k4₄₃ +k4₄₄ +k5₁₅ +k5₁₆ +k5₁₇ +k5₂₂ +k5₂₃ +k5₂₄ +k5₄₈+k6₁₁ +k6₁₃ +k6₁₄ +k6₁₅ +k6₁₆ +k6₁₇ +k6₃₁ +k6₃₃ +k6₃₄ +k6₃₅ +k6₃₆ +k6₃₇

0×00=k4₂₁ +k4₃₁ +k4₄₅ +k5₂₅ +k6₁₅ +k6₃₅

0×00=k4₂₂ +k4₃₂ +k4₄₆ +k5₂₆ +k6₁₆ +k6₃₆

0×00=k4₂₃ +k4₃₃ +k4₄₇ +k5₂₇ +k6₁₇ +k6₃₇

0×00=k4₂₄ +k4₃₄ +k4₄₈ +k5₂₈ +k6₁₈ +k6₃₈

0×00=k4₂₅ +k4₃₅ +k4₄₁ +k5₂₁ +k6₁₁ +k6₃₂

0×00=k4₂₈ +k4₃₆ +k4₄₂ +k5₂₂ +k6₁₂ +k6₃₂

0×00=k4₂₇ +k4₃₇ +k4₄₃ +k5₂₃ +k6₁₃ +k6₃₃

0×00=k4₂₈ +k4₃₈ +k4₄₄ +k5₂₄ +k6₁₄ +k6₃₄

0×7b=k4₄₁ +k4₄₃ +k5₁₅ +k5₁₇ +k5₁₈ +k5₂₁ +k5₂₃ +k5₄₅ +k6₁₁ +k6₁₃ +k6₁₆+k6₁₇ +k6₁₈ +k6₃₁ +k6₃₃ +k6₃₆ +k6₃₇ +k6₃₈ +k6₄₅

0×1b=k4₄₂ +k4₄₃ +k4₄₄ +k5₁₈ +k5₂₂ +k5₂₃ +k5₂₄ +k5₄₅ +k5₄₈ +k6₁₂ +k6₁₃+k6₁₄ +k6₁₆ +k6₃₂ +k6₃₃ +k6₃₄ +k6₃₆ +k6₄₅ +k6₄₈

0×bc=k4₄₃ +k4₄₄ +k5₁₇ +k5₂₃ +k5₂₄ +k5₄₆ +k5₄₇ +k6₁₃ +k6₁₄ +k6₁₇ +k6₃₃+k6₃₄ +k6₃₇ +k6₄₆ +k6₄₇

0×53=k4₄₄ +k5₁₅ +k5₁₆ +k5₂₄ +k5₄₅ +k5₄₇ +k5₄₈ +k6₁₁ +k6₁₈ +k6₁₈ +k6₃₄+k6₃₅ +k6₃₈ +k6₄₅ +k6₄₇ +k6₄₈

0×79=k4₄₅ +k4₄₆ +k4₄₇ +k5₁₃ +k5₂₃ +k5₂₆ +k5₂₇ +k5₄₁ +k6₁₃ +k6₁₅ +k6₁₆+k6₁₇ 450 k6₃₃ +k6₃₅ +k6₃₆ +k6₃₇ +k6₄₁

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0×fb=k4₄₇ +k4₄₈ +k5₁₂ +k5₁₃ +k5₂₇ +k5₂₈ +k5₄₁ +k5₄₄ +k6₁₂ +k6₁₃ +k6₁₇+k6₁₈ +k6₃₂ +k6₃₃ +k6₃₇ +k6₃₈ +k6₄₁ +k6₄₄

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0×bb=k7₂₄ +k7₃₄ +k7₄₂ +k7₄₄ +k8₁₂ +k8₁₄ +k8₂₁ +k8₂₂ +k8₂₄ +k8₂₆ +k8₂₇+k8₂₈ +k8₃₁ +k8₃₅ +k8₃₇ +k8₃₈

0×e6=k7₃₅ +k7₃₅ +k7₄₅ +k7₄₇ +k7₄₈ +k8₁₅ +k8₁₇ +k8₁₈ +k8₂₂ +k8₂₃ +k8₂₅+k8₂₇ +k8₃₂ +k8₃₃ +k8₃₈ +k8₃₄ 'k8₃₅ +k8₃₇

0×5e=k7₂₈ +k7₃₆ +k7₄₅ +k7₄₇ +k7₄₈ +k8₁₆ +k8₁₇ +k8₁₈ +k8₂₁ +k8₂₃ +k8₂₄+k8₂₅ +k8₂₆ +k8₃₁ +k8₃₃ k8₃₄ +k8₃₅ +k8₃₇

0×77=k7₂₇ +k7₃₇ +k7₄₇ +k7₄₈ +k8₁₇ +k8₁₈ +k8₂₂ +k8₂₄ +k8₂₅ +k8₂₆ +k8₂₇+k8₃₂ +k8₃₄ +k8₃₅ +k8₃₆ k8₃₈

0×42=k7₃₅ +k7₃₈ +k7₄₅ +k7₄₈ +k7₄₇ +k8₁₅ +k8₁₆ +k8₁₇ +k8₂₁ +k8₂₂ +k8₂₆+k8₂₈ +k8₃₁ +k8₃₂ +k8₂₅₃ +k8₃₇ +k8₃₈

0×78=k7₄₁ +k7₄₅ +k7₄₆ +k7₄₈ +k8₁₁ +k8₁₅ +k8₁₆ +k8₁₈ +k8₂₂ +k8₂₄ +k8₂₆+k8₃₁ +k8₃₂ +k8₃₄ +k8₃₅ +k8₃₈ +k8₄₁ +k9₁₁ +k9₂₁

0×85=k7₄₁ +k7₄₅ +k7₄₆ +k7₄₇ +k8₁₂ +k8₁₆ +k8₁₇ +k8₂₁ +k8₂₃ +k8₂₇ +k8₃₁+k8₃₂ +k8₃₃ +k8₃₄ +k8₃₅ +k8₄₂ +k9₁₂ +k9₂₂

0×16=k7₄₃ +k7₄₄ +k7₄₆ +k7₄₇ +k7₄₈ +k8₁₅ +k8₁₅ +k8₂₁ +k8₂₂ +k8₂₄ +k8₂₅+k8₂₈ +k8₃₁ +k8₃₂ +k8₃₃ +k8₃₄ +k8₃₅ +k8₃₇ +k8₄₃ +k8₁₃ +k8₂₃

0×24=k7₄₄ +k7₄₅ +k7₄₆ +k8₁₄ +k8₁₅ +k8₁₇ +k8₂₁ +k8₂₃ +k8₂₄ +k8₂₅ +k8₃₇+k8₄₄ +k9₁₄ +k9₂₄

0×7c=k7₄₅ +k7₄₇ +k7₄₈ +k8₁₅ +k8₁₇ +k8₁₈ +k8₁₈ +k8₂₁ +k8₂₂ +k8₂₅ +k8₂₇+k8₂₈ +k8₃₁ +k8₃₂ +k8₃₅ +k8₄₈ +k8₁₃ +k9₁₆ +k9₂₃ +k9₂₈

0×42=k7₄₆ +k7₄₇ +k8₁₆ +k8₁₇ +k8₂₁ +k8₂₄ +k8₂₅ +k8₂₆ +k8₂₇ +k8₂₈ +k8₃₁+k8₃₄ +k8₃₅ +k8₄₂ +k8₄₅ k9₁₂ +k9₂₂ +k9₂₅

0×5d=k7₄₇ +k7₄₈ +k8₁₇ +k8₁₉ +k8₂₁ +k8₂₂ +k8₂₃ +k8₂₄ +k8₂₅ +k8₂₆ +k8₃₁+k8₃₂ +k8₃₃ +k8₃₄ +k8₃₆ k8₃₇ +k8₄₁ k8₄₂ +k8₄₄ +k8₄₅ +k8₄₇ +k9₁₁ +k9₁₄+k9₂₂ +k9₂₄ +k9₂₅ k9₂₇

0×56=k7₄₈ +k8₁₈ +k8₂₂ +k8₂₃ +k8₂₄ +k8₂₅ +k8₂₆ +k8₃₂ +k8₃₃ +k8₂₄ +k8₃₅+k8₃₆ +k8₃₈ +k8₄₁ +k8₄₂ +k8₄₅ +k8₄₆ +k8₄₈ +k9₁₁ +k9₁₂ +k9₁₅ +k9₁₆ +k9₁₈+k9₂₁ +k9₂₂ +k9₂₃ +k9₂₅ +k9₂₆ +k9₂₃ [formula 39]

[0144] Here, the following equation holds true.

rank(M*_(KH))=N_(m)-N_(r)   [formula 40]

[0145] Thus, the above 168 linear-relation equations are linear-relationequations independent of each other. It is therefore obvious that(2¹⁶⁸-1) linear-relation equations obtained from linear concatenation ofany of the 168 equations on the GF(2) hold true. If the number of suchlinear-relation equations is large, it is feared that a new attack thatthe designer of the encryption method is not aware of is brought about.For this reason, the total number of linear-relation equations obtainedby adoption of the method described above can be used as an indicatorfor the evaluation of the encryption level.

[0146] The present invention has been described in detail by referringto the specific embodiments. It is obvious, however, that a personskilled in the art is capable of correcting and modifying theembodiments within a range not deviating from the principle of thepresent invention. That is to say, the embodiments are explained onlyfor the purpose of disclosing the present invention and not to beinterpreted as limitations imposed on the present invention. The scopeof the present invention should thus be determined by referring toclaims appended at the end of this specification.

[0147] It is to be noted that the series of processes explained in thisspecification can be carried out by using hardware, software or acombination of hardware and software. In the case of software used as anexecution means, a program prescribing the series of processes isexecuted. The program is installed in advance in a memory employed in acomputer including embedded special hardware or a general-purposecomputer capable or carrying out various kinds of processing. Typically,the program is recorded in advance in a recording medium embedded in thecomputer. Examples of the embedded recording medium are a hard disc or aROM (Read Only Memory).

[0148] As an alternative, the program is stored (or recorded) in advancein a removable recording medium temporarily of permanently. Examples ofthe removable recording medium are a flexible disc, a CD-ROM (CompactDisc Read Only Memory), an MO (Magneto-optical) disc, a DVD (DigitalVersatile Disc), a magnetic disc and a semiconductor memory. Then, theprogram recorded on the removable recording medium is presented to theuser as the so-called package software. The program is then installed inthe computer from the removable recording medium described above.

[0149] It is to be noted, however, that the program can also bedownloaded to the computer from a download site by a wirelesscommunication or by a wire communication through a network instead ofbeing presented to the user by using a removable recording medium.Examples of the network are a LAN (Local Area Network) and the Internet.The computer includes functions to receive the downloaded program andinstall the received program in the embedded recording medium such as ahard disc.

[0150] It is to be noted that the various kinds of processing describedin this specification can be carried out not only sequentially inaccordance with a predetermined sequence but also concurrently orindividually in accordance with the processing capacity of the apparatusfor performing the processing or in accordance with the necessity.

[0151] As described above, in accordance with the configuration of thepresent invention, it is possible to comprehend all equations expressinglinear relations among round keys in the common-key block encryptionmethod without regard to the complexity of key scheduling and possibleto evaluate the encryption level of the common-key block encryptionmethod on the basis of the derived equations expressing linear relationsamong round keys.

[0152] In addition, in accordance with the configuration of the presentinvention, the key-scheduling part algorithm, which is one of encryptionalgorithms, is expressed in terms of equations represented by vectorsand a matrix and, then, non-linear transformation output values andinitial values are eliminated from the matricial equation by carryingout a unitary transformation process in order to find all equationsexpressing linear relations among round keys. If the relations among theround keys are simple dependence relations, the number of true roundkeys decreases. Thus, the designer of the encryption method needs to usecaution so as to prevent a large number of such relation equations fromexisting. In accordance with the method provided by the presentinvention, the level of encryption keys is evaluated for the purpose ofreducing the number of equations expressing linear relations among roundkeys. As a result, a safer encryption method can be designed.

What is claimed is:
 1. An encryption level indicator calculation methodbased on an encryption processing algorithm and composed of: a step ofsetting a common key block encryption processing algorithm, which is toserve as said encryption processing algorithm to be used as the base ofsaid encryption level indicator calculation method, has a key-schedulingpart comprising a linear transformation part and a non-lineartransformation part and includes: a sub-step of generating initialvalues U_(i) (where i=1, 2 and so on) from a master key; a sub-step ofcalculating intermediate values Z_(i) ⁽⁰⁾ (where i=1, 2 and so on) fromsaid initial values U_(i) (where i=1, 2 and so on); a plurality ofsub-steps of calculating intermediate values Z_(i) ^((r)) (where i=1, 2and so on) from intermediate values Z_(i) ^((r-1)) (where i=1, 2 and soon); a sub-step of calculating said non-linear transformation partoutputs V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) fromsaid intermediate values Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2and so on) and said initial values U_(i) (where i=1, 2 and so on); and asub-step of calculating round keys K_(i) ^((r)) (where i=1, 2 and so onand r=1, 2 and so on) from said intermediate values Z_(i) ^((r)) (wherei=1, 2 and so on and r=1, 2 and so on) and said non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on); a step of eliminating said intermediate values Z_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on) serving asvariables so that said round keys K_(i) ^((r)) (where i=1, 2 and so onand r=1, 2 and so on) can be expressed as a linear combination of saidinitial values U_(i) (where i=1, 2 and so on) and said non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on); a step of transforming said linear combination into asimultaneous linear equation completing transposition of terms and,thus, consisting of only terms of said initial values U_(i) (where i=1,2 and so on) and said non-linear transformation part outputs V_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on) on the right-handside of said equation; a step of transforming said simultaneous linearequation into a matricial equation; a step of multiplying both theleft-hand and right-hand sides of said matricial equation by arow-deform unitary matrix deforming a matrix on the right-hand side ofsaid matricial equation obtained as a result of transformation into astep matrix from the left; a step of creating a new matrix consisting oflowest N rows of a matrix on the left-hand side of said matricialequation obtained as a result of transformation where N is a numberobtained as a result of subtracting the rank value of said step matrixfrom the number of rows in said step matrix; and a step of finding Nlinear-relation equations by multiplying a column vector consisting ofsaid round keys K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and soon) as elements by said new matrix generated at said preceding step,.where: symbol U_(i) (where i=1, 2 and so on) denotes an initial value ofsaid key-scheduling part; symbol Z_(i) ^((r)) (where i=1, 2 and so onand r=1, 2 and so on) denotes an intermediate value of saidkey-scheduling part; symbol V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on) denotes an output of said non-linear transformationpart; and symbol K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and soon) denotes a round key calculated from said intermediate values Z_(i)(where i=1, 2 and so on).
 2. A program to be executed as a computerprogram in carrying out an encryption level indicator calculationprocess based on an encryption processing algorithm and composed of: astep of setting a common key block encryption processing algorithm,which is to serve as said encryption processing algorithm to be used asthe base of said encryption level indicator calculation process andincludes: a sub-step of generating initial values U_(i) (where i=1, 2and so on) from a master key; a sub-step of calculating intermediatevalues Z_(i) ⁽⁰⁾ (where i=1, 2 and so on) from said initial values U_(i)(where i=1, 2 and so on); a plurality of sub-steps of calculatingintermediate values Z_(i) ^((r)) (where i=1, 2 and so on) fromintermediate values Z_(i) ^((r-l)) (where i=1, 2 and so on); a sub-stepof calculating said non-linear transformation part outputs V_(i) ^((r))(where i=1, 2 and so on and r=1, 2 and so on) from said intermediatevalues Z_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) andsaid initial values U_(i) (where i=1, 2 and so on); and a sub-step ofcalculating round keys K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2and so on) from said intermediate values Z_(i) ^((r)) (where i=1, 2 andso on and r=1, 2 and so on) and said non-linear transformation partoutputs V_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on); astep of eliminating said intermediate values Z_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) serving as variables so that said roundkeys K_(i) ^((r)) (where i=1, 2 and so on and r=1, 2 and so on) can beexpressed as a linear combination of said initial values U_(i) (wherei=1, 2 and so on) and said non-linear transformation part outputs V_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on); a step oftransforming said linear combination into a simultaneous linear equationcompleting transposition of terms and, thus, consisting of only terms ofsaid initial values U_(i) (where i=1, 2 and so on) and said non-lineartransformation part outputs V_(i) ^((r)) (where i=1, 2 and so on andr=1, 2 and so on) on the right-hand side of said equation; a step oftransforming said simultaneous linear equation into a matricialequation; a step of multiplying both the left-hand and right-hand sidesof said matricial equation by a row-deform unitary matrix deforming amatrix on the right-hand side of said matricial equation obtained as aresult of transformation into a step matrix from the left; a step ofcreating a new matrix consisting of lowest N rows of a matrix on theleft-hand side of said matricial equation obtained as a result oftransformation where N is a number obtained as a result of subtractingthe rank value of said step matrix from the number of rows in said stepmatrix; and a step of finding N linear-relation equations by multiplyinga column vector consisting of said round keys K_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) as elements by said new matrix generatedat said preceding step, where: symbol U_(i) (where i=1, 2 and so on)denotes an initial value of said key-scheduling part; symbol Z_(i)^((r)) (where i=1, 2 and so on and r=1, 2 and so on) denotes anintermediate value of said key-scheduling part; symbol V_(i) ^((r))(where i=1, 2 and so on and r=1, 2 and so on) denotes an output of saidnon-linear transformation part; and symbol K_(i) ^((r)) (where i=1, 2and so on and r=1, 2 and so on) denotes a round key calculated from saidintermediate values Z_(i) (where i=1, 2 and so on).